There is very little optical performance difference between these two lenses when both are mounted on an APS-C body. The only difference is the range of focal lengths each offers when both are used on an APS-C body. I personally don't think it is worth an additional $350+ expense (assuming you can sell your current lens for a good price) to move from the 55-250mm to 70-300mm when the EF-S 55-250mm f/4-5.6 that you already own costs less than half the price of the EF 70-300mm f/4-5.6. In fact at 100mm and below the 55-250 is sharper at widest aperture than the 70-300. At 135mm and above the 70-300 is slightly sharper than the 55-250 when both are wide open mounted on an APS-C Body. The 70-300s greatest advantage in sharpness comes at the long end of both lenses focal range. Here's the side-by-side comparison at 100mm and f/5. You can change the focal length and aperture of each lens to compare each at other settings.
One of the factors to consider when shooting moving objects with a telephoto lens is the lens' maximum aperture: the wider the aperture the 'faster' the lens is said to be because you can select a faster shutter speed for the same ISO. A wider maximum aperture also gives your autofocus system more light to work with, which usually means faster focusing. Neither of these lenses is great in that regard. A 70-200 constant aperture zoom gives you the ability to use faster shutter speeds at the longer focal lengths and that is where you need them most. Not only are you fighting the motion of your subject which is magnified with longer focal lengths, but you must also consider that the narrower the FoV of the longer lens, the more the same amount of camera motion will blur your pictures. At 200mm an f/4 lens allows you to use a shutter speed twice as fast as an f/5.6 lens, and an f/2.8 lens allows a shutter speed four times faster. The difference between 1/250 second and 1/500 second shooting athletes from the sideline is significant, and the difference between 1/250 second and 1/1000 second is huge.
If you want significant improvement over the EF 55-250mm f/4-5.6 in terms of image quality then you need to consider one of the constant aperture 70-200 "L" lenses. Not only are they faster than the 55-250 or the 70-300, but they are sharper as well. Here's an answer that runs down the differences between each version of the Canon 70-200mm lenses as well as a few third party options. Here is an answer to another question directly comparing the EF-S 55-250mm f/4-.56 IS II and the EF 70-300mm f/4-5.6 IS. If you want considerably more reach than the 55-250 provides, you might want to consider a lens such as the EF 100-400mm f/4.5-5.6 L IS. See also this answer to Moving up to the next lens from 55-200. And here is a side-by-side comparison of the 70-300 and the EF 70-200mm f/4 L at 200mm. At f/4.5 the 70-200 is noticeably sharper than the 70-300 at f/5. DxO Mark has only tested the 70-300 on Full Frame bodies, but here are the 70-300 and 70-200 f/4 compared on the 1Ds Mark III. If you click 'Measurements-->Sharpness-->Profiles' the edge of the frame for a crop sensor body will be at the 62% field position. If you select 200mm for both lenses and f/5.6 (the widest common aperture tested) you can see the significant difference in sharpness. Also be sure to click 300mm to see the loss of sharpness in the center past 200mm for the 70-300.
In terms other than technical optical performance, the 55-250 weighs in at 13.8 oz (390g), the 70-300 weighs 22.2 oz. (630g). While the 70-300 has USM AF, it is Micro USM, not Ring USM, and is not greatly different from the 55-250 in AF speed. It is a little quieter, which often makes a lens seem to be faster. The mount ring on the 55-250 is plastic, the 70-300 has a metal mount ring.
If you eventually see yourself moving up to an "L" series level telephoto zoom, I'd advise you stay with the 55-250 for now and save the extra money towards the eventual purchase of the lens you ultimately want. If you have no plans to ever move past the 70-300, then it might be worth the cost of upgrading in terms of durability and the very marginal increase in performance.
The following is an attempt to illustrate the difference the extra aperture and higher resolution of the Canon "L" series 70-200 zoom lenses offer over lenses such as the 55-250 or the 70-300. I tried to find photos taken near in time to each other with the same camera body but the two different lenses in order to remove the variables of a better camera body or an increase in my own skill level.
Here's a sample taken with the EF-S 55-250mm f/4-5.6 IS II mounted on a 50D in bright daylight. The dragonfly was using its wings to hover so they were moving rather rapidly.
ISO 200, 1/500 sec, f/8, 250mm. Uncropped at 4752x3168 before being re-sized to 640x420.
And a 100% crop of the subject. It is a little soft, but not bad. Remember, this is at f/8.
Here's a sample taken with the EF 70-200mm f/2.8 L IS II mounted on the same 50D under the lights at night in a decently lit high school football stadium (A lot less light to work with than the shot above). The subject was moving toward the camera position and spinning the flag at the time the photo was taken. ISO 800, 1/200 second, f/2.8, 120mm. Cropped from 4752x3168 to 4266x2844 before being re-sized to 640x420.
And a 100% crop. You can see the weave of the fabric, even with moderate NR applied due to the low light! And this is at f/2.8. Three stop stops wider than the above photo taken at f/8.
With the 55-250 or the 70-300, the lens would have been at f/5 at the focal length used for the second shot. To get the same exposure the Tv would have been 1/60 second, and the camera's Phase Detection AF system would have had to work much harder to achieve focus through the narrower lens, if it could have achieved focus at all. Even if it focused just as accurately in the same amount of time, the 3.33x longer Tv would have meant the hands and flag would have moved 3.3 times as far during the exposure and thus they would have been blurred even more than they were at 1/200 second.
Then there's always the odd occasion when you want to photograph a pilot at an air show flying like his hair's on fire at 200 mph and 3/4 mile away - at night!
Canon 7D + EF 70-200mm f/2.8 L IS II. ISO 6400 (and the necessary NR), 1/60 second, f/2.8. Cropped from 5184x3456 to 3565x2377 before resizing to 1536x1024 for web viewing.