I am on a tight budget and chose to use GIMP for editing since it is free.
What important photographic post-processing features am I missing from Photoshop?
For photos? Not too much, actually. GIMP lacks automatic HDR processing. It doesn't have adjustment layers - although you don't need those too much for photos. Photoshop's Hue\Saturation dialog is superior. Photoshop CS5 has content-aware fill, which GIMP lacks, but there's a GIMP plugin called Resynth that does about the same thing:http://www.logarithmic.net/pfh/resynthesizer
Some pretty good art has been done in GIMP. (My snow photomanip, for instance) It's more about the artist's skill than the tools he\she uses.
Gimp is great, but it's not without some shortcomings. Photoshop is a big-time commercial project with a lot of funding, and while Gimp's development community is awesome, there are a lot of areas which could be useful to photographers where Photoshop is ahead. I've tried to enumerate these here as fairly and as realistically as I can.
Gimp is under active development, and the "roadmap" can be found at http://wiki.gimp.org/index.php/GIMP_Roadmap. This is useful for getting an idea of what shortcomings will be rectified soon, and what else is coming in the near future.
Gimp 2.8 was released in May 2012, and it features a number of significant improvements to the user interface, most notably a single-window mode. This is not a static area — read this January 2012 blog post for an idea of work in progress from Gimp's interaction-design team. As a long-time Linux user actually happy with the multi-window paradigm, I thought I'd hate the change, but it's actually done with a lot of thought, not just everything glommed together. This is very good work. There's also multi-column dock windows, and other refinements to make better use of space and to make the UI more friendly to people working with graphics tablets. If you haven't used the software in a while, give it a try and see what you think.
There's still some UI things which could use serious work. Frequently-used items are buried too far in disorganized menus, and while it's easy to remap keyboard shortcuts, there's no good way to tailor the menus for photographic tasks specifically, or to move favorite menu items to shortcut bars. That means more clicking than I'd really like, and it means that some great features are hard to discover. I think this will get better in time too, but the big change to single-window had to happen first.
There are a number of other things like RAW development and lens profile correction which Photoshop does and which are not handled well in Gimp but which are covered by other open source tools (like Hugin, RawTherapee, and Darktable).
This may seem biased/unfair, but GIMP has awful usability.
Disclaimer: i have used both, although recently i use photoshop much more.
In terms of functionality, GIMP doesn't tread much behind photoshop (for simple photomanips and adjustments - i don't do HDR), but layer manipulation and general use are tortuous (IMHO). I never couldn't do a task with GIMP, but i end up searching online how to do it, whereas in Photoshop i tend to find stuff on the menus or figure it out by myself.
And yes, i am aware that 600€ is quite a lot to pay for a small hobbyist.
EDIT: (in reply to the comments) Yes, awful is a bit vague.
I wasn't really referring to the panel approach, it's slightly confusing but not as serious as the layer manipulation controls in an image, selecting, dragging, enlarging, etc.
I didn't mean to bash, i learned the basics in GIMP, it's selection editing tools are fairly evident, but i didn't use many of GIMP's functionality (3+ layer composites, filters, color corrections) because i didn't know the name of what i was trying to accomplish, or simply because fell into dead ends (not being able to do something, not figuring it out quickly, getting frustrated, giving up) Only after transitioning to photoshop, and learning those tasks, did i know what to look for in GIMP.
I am not sure if i made myself clear, i am referring to findability vs discoverability (http://maadmob.net/donna/blog/2005/findability-vs-discoverability). I defend that photoshop is a very much better learning tool because it enables its users to discover functionality that they didn't even know they could do in the first place.
I suppose you COULD do them almost all in GIMP, after you know what to search for, and investing the time in learning it, but photoshop allowed ME to learn almost all that i know with almost no research at all.
Usability may seem something rather trivial or useless, but it really isn't. It is very complicated to get right, and is never ideal for everyone. But Adobe has obviously put a lot of effort into making Photoshop usable, and it shows. Also, i am aware of the difficulty for FOSS to engage in user testing and usability evaluation, because of the nature of the projects development structure (many developers, far apart, functionality oriented).
One item I don't see mentioned in other answers is performance. Especially on a Mac. Lightroom and Photoshop both crunch through adjustments and work MUCH faster than the Gimp does.
One key difference is not in the product itself, but its development. There are about two developers working on Gimp, and as a result, new features take a long time to be production ready. They 16-bit GEGL engine has been in progress for a couple of years and is not yet released.
I don't think it is fair to say that the Gimp UI is terrible, but it is very different from Photoshop, and its not nearly as polished.
I used to use Gimp all the time, and was very happy with it. But these days I use Aperture. I don't need pixel editing, the basics of crop/rotate and some exposure controls are what I need.
For those looking for an open-source package that is closer in spirit Aperture or LightRoom, look at Darktable.
When I process my photos, apart from color correction the two functions I most need are Smart Sharpen and Reduce Noise, and unfortunately both are missing from GIMP.
My understanding is that the algorithms used by Photoshop are proprietary and trade secrets, so you can't just implement them to GIMP, you'd need to reverse engineer or reinvent them yourself.
Noise reduction is in essence about removing unwanted details without removing wanted details and as such it's black art. However, you can work around this by shooting with lower ISO - using tripod or off-camera flash if necessary.
Smart sharpen is harder to replace. It basically makes the image look like the original version was slightly blurred version of the resulting image, instead of adding halos around high-contrast edges like most trivial sharpening algorithms do.
As the other answerers have pointed out, GIMP misses certain features that you can find in photoshop. But then you are not limited to using only GIMP. I use the following free of charge programs:
dcraw allows you to have full access to your raw files, you can e.g. work with the raw data before any demosaicing is done. ImageJ allows you to easily manipulate the raw data of your picture in exactly the way you want it, unlike higher end programs like GIMP. ImageMagick allows you to easily manipulate images via command line instructions, it allows you to perform batch operations. Hugin is a panorama stitcher which includes the programs "align_image_stack" and "enfuse" which you can use to align images and compose HDR images, respectively.
In some cases I have used all these programs to do post processing to produce one picture. E.g. to sharpen an image I've used dcraw to extract the slightly blurred raw image of a star before any demosaicing. That image of a star the served as the point spread function that I used to deconvolve the image with using an imageJ plugin (the reason why you need the image of the star before demosaicing is due to the severe demosaicing artifacts when the brightness changes changes significantly over the range of just a few pixels). But this requires working in linear colorspace, and for that I used dcraw to convert the image to a 16 bit linear tiff file.
I did this for several pictures of the same scene and then I used the "align_image_stack" program from Hugin to align the images. Then using imagemagick, I could compute the "maximum" and "minimum" of the images (i.e. the images obtained by taking the maximum or minimum grey values of each pixel in the aligned images), and then you can computed the average of all the images where you subtract the maximum and the minimimum (this reduces noise by averaging as well as removing the outliers). Then I did this for different exposures and the results of the different exposures could be combined into a HDR image using the enfuse program (before that I had to converted the images to sRGB using ImageMagick). Finally with GIMP I could make some final adjustments and convert the tiff file to jpeg.
It may well be the case that with photoshop you could do all of this work, but I doubt if you could work smoothly with any single program. Some programs are more suitable for doing lower level processing while others are better at doing higher end photo editing.