A FF (Full Frame)sensor has about a 50% advantage in resolution compared to an APSC sensor for equal sensor pixel density. For meaningful copmpaarison consider the case where the identical FF lens with the same settings (focal length, aperture) is used to photograph the same scene using a FF and an APSC camera, such that the identical scene area is reproduced in the out-of-camera image in each case. In this scenario the FF sensor uses essentially the whole of the lens area and the APSC camera uses half of the lens area, mostly in the centre of the lens. To achieve this comparative result with the same focal length settings in each case the FF user must be proportionally closer to the subject. Adjusting focal length to equalise image sizes invalidates the comparison.
If lens sharpness / quality / contrast / MTF gets progressively worse on average towards the edges compared to the centre, as is the case with all lenses affordable by mere mortals, then a FF sensor is more affected than an APSC sensor, as the FF sensor uses the whole lens image and the APSC sensor uses the higher quality middle portion.
Whether the FF's ~= +40% dpi advantage over the APSC offsets the degradation in lens quality at the edges depends on the lens parameters and aperture and focal length settings. With extremely high quality high cost lenses the FF sensor will be sharper at all locations under all conditions. With more ordinary lenses a FF sensor will be substantially sharper in the centre and less sharp at the edges than the APSC in absolute terms, and especially so in the corners.
As a lens is stopped down the image size remains the same but the outer portions of the lens are not used. This means that the APSC "centre of lens advantage" decreases as aperture gets smaller and a FF sensor should be sharper across the range at small apertures.
The above summary can be confirmed by looking at the Tamron FF SP 70-300mm f/4-5.6 MTF charts at the end of this post. In these Tamrom show the results for APSC & Full Frame sensors and you can scale the curves by whatever crop factor applies. It can be seen (as expected) that at the centre the Full Frame is clearly superior, while at the corners or edges the result varies with lens setting and in some cases, especially at large apertures, the APSC results will be superior across a significant portion of the image.
In the diagram below from here
The black outer circle represents the image area represents the image formed by a FF lens. The blue rectangle = the FF sensor and is almost touching the image circle. Clearly the diagonal corners of the sensor are a lot closer to the image edge than the outer extents of the vertical or horizontal axes are.
The green rectangles = the APSC sensor area are very comfortably inside the FF lens' image area and while the diagonal corners are closer to the corners than the extents of the vertical or horizontal axes are.
Assume that the FF sensor is exactly twice the APSC sensor area and that they both have equal pixel density per area, so that the FF sensor has twice has many pixels. The linear pixel density if square troot of two as great or about 41% higher for the FF sensor. ie the FF sensor has 40% more sensor cells in a straight line to assist it in obtaining the best possible line pairs per mm (or per inch).
For a lens that is equally good across the whole lens area this gives a clear advantage to the FF camera. Very expensive high quality lenses are therefore liable to give a substantially improved result with a FF sensor.
When using a more typical FF lenses on either a Fullframe or APSC camera (same lens in boith cases) with the same subject area filling the frame, an APSC sensor is liable to give a superior result when the lens is "wide open" or at the low focal length end of its range.
Real world lenses tend to have inferior performance towards the edges compared to the centre with results usually but not always increasing with distance from centre. As the FF sensor is using portions of the lens firther from the centre than the APSC sensor is it has its resolution advaantage opposed by lens quality disadvantages. The relative diffierence betwwen the rea os the lens used by the APSC sensor and the FF sensor govern whether the FF gains or loses overall due to its superior resolution.
Also, if lens quality falls with distance from centre, the FF will tend to have a greater variation in vertical to horizontal edge sharpness than an APSC sensor using the same lens, because the ratio of diagonal to horizontal distances as a fraction of lens image diameter are larger for a FF than an APSC sensor. This means that is a lens softens progressively towards the edges the diagonal edges (= corners) will be relatively softer than the middle or the horizontal axes edges than with an APSC sensor. (The same applies to vertical axes edge to corner distances and softness.
When a lens is stopped down somewhat or zoomed in somewhat the FF sensor will benefit more with a typical lens and is liable to about equal results with a reasonable quality lens and superior results with a very good to excellent quality lens.
ie if you can afford Zeiss lenses then use a FF camera :-)
A Full Frame camera *with the same lens as a half frame will usually (but not always) produce a SOFTER image.*
To allow reasonable comparison assume a FF camera with exactly twice the sensor area of an "APSC" camera and equal pixel density per sensor area, so twice the megapixels. eg a 24 Mp FF and a 12 Mp APSC sensor.
For the cameras to be using the same lens, which is what was asked, the lens must be a FF lens. The FF camera will use essentially the whole lens area (by design) and the APSC camera will use a small more central area of the lens. While it is technically possible to make a lens which has close to equal performance across the whole lens area, in practice lenses which mere mortals can afford tend to be softer towards the edges. The FF camera must deal with these edges and include them in the image whereas the APSC camera has them automatically excluded.
If a photos is taken from the same position with the same lens and with the same lens settings in each case the APSC image will be of 50% of the area that is seen in the FF image as the APSC sensor is 50% of the area of the FF sensor and it is being exposed to the same optical image by the same lens.) If the FF image is cropped to the same as the APSC image then you have identical image content being processed by equal sensor area and the results are identical for cameras with equal pixel density per sensor area. The results are identical.
If instead, the FF camera image is recomposed by either changing the lens settings (eg focal length increase by a factor the crop factor) change) or by moving closer so identical images areas are produced, the FF camera will now have the same image on twice the sensor area. Lines per inch are improved by a factor of 1.414 (because, as the sensor is 2x area, the linear dimensions are square root of 2 larger for the same sensor aspect ratio). This taken in isolation would improve the sharpness. However, the whole lens is now being used. If the MTF (modulation transfer function = measure of lens quality / contrast resolving power / sharpness) is worse by a factor of ~ 1.4 in any location then the lens will be less sharp in those area. So, in all locations it will be liable to be more sharp due to sensor resolution gains but at the edges many lenses will be worse due to MTF dropoff. Note that MTF variation differs (often widely) at differing apertures and focal length settings (for zooms) and certainly between different lenses.
The diagrams below, from here were chosen NOT to cherry pick my point, but simply as the first useful one I found with a web search, and demonstrate the above point. The lens is not an overly marvellous one and is a "DX" (APSC) lens but illustrates the point well enough - probably better than some due to it not being an overly expensive lens. While it is a DX lens, it is legitimate for this comparison to think of it as a FF lens with the APSC sensor using the centre to middle ranges.
At f/3.5 and 18mm the differences between centre/border/extreme border are so pronounced that when used in FF you might think someone had used purposeful softening around the edges.
At f/5.6 and 18mm the border with our example sensors is perhaps just sharper with FF and extreme border is still softer.
By f/8 and 18mm extreme border is still just down on ff compared to APSC.
By f/11 and 18mm the lens as a while is getting softer (still very good in the middle) and the MTF losses even on the extreme border are more than made up for by the FF's lpi gain.
ie with this lens, at 18mm focal length and large apertures the centre would be sharper on FF but the edges would be noticeably softer and by f/11 it would be much sharper in the middle and somewhat sharper at the extreme borders.
The following graphs show results at increasing focal length. At 35mm the APSC is still sharper at the edges at large aperture and by 80mm and above, where the FF is not using the lens edges, the FF is clearly superior.
Here's an example where Tamron have done the work for me. From here
This is for a Tamron FF SP 70-300mm F/4-5.6 Di VC YSD model A005 lens (!).
Graph curve colurs can confuse.
A given lp/mm count has a red curve (radial) and a blue curve (circumferential).
Tamron very helpfuilly show APSC and Full Frame cutoff lines.
Looking at the right hand graph - at 300mm f/5.6 the FF wins easily on radial results.
At 10 line pairs/mm the response is close to straight line radially and not much worse at 30 line pairs/mm. In fact at 30 lp/mm it's superior radially for FF than for APSC before the sensor resolution gain is allowed for.
Circumferentially (blue lines) the FF fades badly compared to the APSC - so much that the APSC will be superior even allowing for sensor increase. Reading Tamron's text they suggest that 10 lp/mm is a measure of contrast and 30 lp/mm is a measure of sharpness. In practice they are both closely related but that simplification is good enough as a first assessment.
Tamron are saying that for circumferential results at 300mm f/5.6 the lens has better to much better contrast with a FF sensor but will have superior overall sharpness with an APSC sensor. Overall = ???
You'd have to take it out and play, but it's not clear that either FF or APSC will be a certain winner overall.
The left hand graph = 70mm, f/4 is less kind to the FF sensor and the APSC has a clearly visible edge overall for sharpness and is similar for contrast (if you decide you can in fact split these two measures). This is not unexpected with the lens "wide open" and using all the glass in FF mode.
This is because the FF uses all of the lens area and the APSC uses the centre portion. It is hard for a lens maker to maintain equal quality across the lens surface and hardest at the edges. Using the centre of th elens tends to produce a sharper result. In some cases this "rule" is broken and a given lens may work better on a full frame for various reasons, but this is not usually what happens.
Matt and I may aappear to disagree on this point but probably don't. Using the same lens as a reference is necessary for comparisons.
APSC cameras are on average much lower cost than FF cameras and lenses used with them are usually lower cost. This is of course up to the user and some people will buy very high quality high cost lenses and use them on APSC cameras, but in most cases a user will migrate to a FF as they buy 'more expensive glass'. An exception may be sports photographers using Canon systems who use Canon's cropped sensor cameras due to their higher frame rate and features which to some target high ISO high speed photography.
The biggest factors influencing softness are lens quality and aperture.
Almost all lenses produce their maximum sharpness when used at less than full aperture. There are exceptions, but they are rare, and cheaper lenses always benefit from "stopping down". Odds are you used a lens with maximum aperture of around f/3.5 and it may have been used at say f/5.6 in that image - maybe not. With a cheaper lens best results are usually achieved at f/8 or smaller aperture. Initailly the image sharpens as aperture is decreased (larger f number). Somewhere, usually in the f/11 to f/22 range diffraction effects start to soften the image again. Some lenses are starting to diffraction soften at f/11 and the very best may get to around f/22. (Some eg Ansell Adams images are up around f/40 but with large format cameras the 'rules' change.)
If you want a sharo image with a cheaper lens you need to experiment to find its optimum aperture. Also be sure that shutter speed is fast enough to not have motion causing softening due to motion blur.
What were the camera settings for your "soft" image.
Can you provide a web link to some "sharp" images.
Your f/2.8 cat photo MAY be very sharp in the original BUT over a very limited depth of field. DOF is a quite different issue to sharpness. When shooting at f/2.8 you either have all the subject in a very shallow distance range if you want it totally sharp OR you not only accept but usually intend that all except a small band of distances will be out of focus. This effect is usually sought after AND will be more pronounced on a FF camera all else being equal. The effect will be reduced with increasing distance to subject, decreasing aperture (larger f number) and shorter focal length.
The examples that you give from istockphoto MAY be sharp all over as you think but are too small (low resolution) to be sure and have been taken with settings aimed at ensuring subject sharpness overall.
Try taking photos at f/8 and f/16 and see what the result is. When focsing pay special attention to getting focus "spot on". If you have a focus magnifier feature in the camera use it.