First, let's get a bit of nomenclature out of the way:
EF-S means a lens in the Canon EOS system that will only work on cameras with APS-C or smaller sensors (if any smaller than APS-C EOS sensors are ever created). They can not be used on cameras with larger 35mm film sized "full frame" sensors.
55-250mm is the focal length of the lens. On an APS-C camera, a 55-250mm lens gives the same approximate field of view/angle of view that an 88-400mm lens would give on a 35mm film camera or FF digital camera. This only matters if you are familiar with 35mm focal lengths and want to know what the "equivalent" angle of view would be on cameras with a different format size sesnor.
f/4-5.6 refers to the lens' maximum aperture. A lower f-number is a wider aperture which lets more light into the lens. This allows a faster shutter time or lower ISO to be used when desired. Both of these lenses have a maximum aperture of f/4 at 55mm. The maximum aperture gradually narrows at focal lengths between 55m and 250mm. At 250mm the maximum aperture is at f/5.6.
IS Image stabilization is a technology that helps to counteract movement of the camera/lens during the duration of an exposure. It allows using longer shutter times without blur caused by camera motion when the camera is handheld than would otherwise be possible. It does not help with motion blur caused by movement of the subjects in your photos. Only a shorter shutter time can do that.
II The Roman numeral two. It indicates that this is a second, updated version of a previous lens with the same name (other than the addition of the "II"). A subsequent version would be named "III".
STM Stepping motor. Refers to one type of autofocus motor used to move a lens' focusing elements.
There are a couple of key differences between the EF-S 55-250mm F/4-5.6 IS II and the EF-S 55-250mm f/4-5.6 IS STM:
- Two different optical formulae. Even though the names are very similar, the optical formulae of the lenses are not the same. The EF-S 55-250mm f/4-5.6 IS II (2011) is optically identical to the earlier EF-S 55-250mm f/4-5.6 IS (2007). Though not always the case when comparing an original version to the "II" version of a lens from Canon, the differences between the original 55-250mm "IS" and the "IS II" are purely cosmetic other than an updated IS (image stabilization) firmware. These lenses had 12 lens elements in 10 groups. The "STM", on the other hand, has 15 lens elements in 12 groups.
- The way the focusing elements in the lens are moved. The original 55-250mm "IS" and "IS II" had micromotor autofocus. This is the most basic (and cheapest) type of AF motor in Canon EOS lenses. They're good enough, but they are neither as fast nor as quiet as Ultrasonic Motor (USM) AF units found in higher end Canon EOS lenses. The newer "IS STM" uses a newer type of stepping motor to move the lenses focusing elements.
Differences in optical design, of course, can affect the optical performance of the lens. In this case the "IS II" and the "IS STM" are very similar from 55mm to about 100mm. Beyond 100mm the general consensus of most reviewers is that the newer STM lens maintains more or less the same level of "sharpness" all the way to 250mm, while the older "IS II" begins to become increasingly soft as it is used at 135mm and beyond, with the the worst decrease in performance coming at the longest 250mm focal length ("zoomed in all of the way"), particularly at the edges and in the corners of the frame.
The other significant difference in the optical design is that the increased number of lens elements in the "STM" are in the rear of the lens to allow the lens to use internal focus. This allows the "STM" lens to focus without rotating or moving the front of the lens barrel forward/rearward as the focus is adjusted. The front barrel of the "IS II" rotates and moves in/out as the focus is adjusted. The front barrel of the "STM" does move in/out when zoomed, but the barrel does not rotate during zooming. If one is using polarizers or graduated neutral density filters that are affected by lens rotation this can be a distinct advantage as the filter does not require adjustment every time the focus distance is changed.
The STM autofocus motors were designed to be smoother and quieter than micromotor AF. The advantages for shooting video should be obvious: focus can be altered during a shot with smooth transition from one distance to the other while also being quieter and less likely to be picked up by the camera's microphone. Compared to micromotor lenses, the STM lenses tend to be faster focusing as well as smoother and quieter. The higher end USM lenses are optimized for shooting still images. They're also usually fairly quiet, but they place a premium on focusing as fast as possible without worrying about any "jerkiness" between focus positions. The emphasis is to get to the desired focus position as soon as possible so the still photo can be taken NOW.
One additional key difference between most micromotor AF lenses and STM has to do with manual focusing. To manually focus a micromotor lens, the AF/MF switch must be moved from AF (autofocus) to MF (manual focus). The lens is then focused using a mechanical connection between the focusing ring on the lens and the lens elements inside the lens. Moving the focusing ring with the switch set to 'AF' can damage the focusing mechanism. With the 'AF/MF' switch set to 'MF', the lens can be focused even when not connected to a camera.
STM lenses are all 'focus-by-wire'. There is no direct mechanical connection between the focusing ring and the lens' focusing elements. When the focusing ring is turned it sends a signal to the camera which sends a signal back to the lens telling the focusing element to move. The finest focus adjustment that can be made, even in manual focusing, is a single 'step' of the AF motor. This allows 'full-time' manual focusing, even when the 'AF/MF' switch is set to 'AF'. It also requires the lens to be attached to a camera that is powered on to move the focusing mechanism.
(Most USM lenses are full-time manual focus via a mechanical connection between the focusing ring and the USM's ring inside the lens. This allows the lens to be manually focused even when not attached to a camera. There have been a few 'focus-by-wire (only)' USM lenses. As of December 2017 all but one have been discontinued.)
So which is "better?"
It all depends on what you want to do with the lens.
- The "STM" is a slightly better optical performer.
- The "STM" has faster, quieter AF, but does require the 'focus-by-wire' connection to focus in discrete steps.
- The "IS II" can be manually focused without any power from the camera and can be focused in finer differences than the smallest 'step' of the stepper motors in STM lenses.
- The "IS II" is generally cheaper, which allows the money saved to be spent on another lens or piece of lighting gear or other accessories.
For most folks the newer EF-S 55-250mm f/4-5.6 IS STM would be the more desirable choice in all but budgetary considerations. But the older "IS II" is certainly capable of taking good photos almost on par with the newer model.