Both have decent image quality. The Canon STM are mostly very recent designs, often the latest-and-greatest version of the lens with other improvements besides the better AF drive system. For example, the 55-250 STM is more compact, closer focusing than the earlier, non-STM version. It also has a non-rotating front barrel (which makes it easier to use with rotating filters such as polarizers).

So long as it meets reasonable standards, image quality factors really shouldn't matter very much comparing these two lenses because they serve different purposes. You should base your decision more on what focal length range you need and/or what other lenses you already have. If you already have an 18-55mm, for example, the 55-250 might make more sense. If you want a single lens to carry most of the time, the 18-135mm might be a better choice. If you mostly shoot landscapes, portraits, etc. the 18-135mm might be more than enough (or you may want to complement it with a wider lens like the EF-S 10-18mm STM IS or EF 10-22mm USM). If you want to shoot sports or wildlife, you are more likely to want the stonger telephoto "reach" of the 55-250mm.

I use a T3i currently. Started with the basic 18-55 "kit lens" and then bought the 55- 250. These are OK lenses to start with, but I definitely felt an upgrade was worthwhile. My wife now has those on a T2i and is happy (at least for now).

I now have the 18-135 STM and the 70-300 IS, together with the 10-22 ultra-wide zoom as my primary three lenses. I have other more esoteric things, but I find that a much better combination than the 18-55 and 55-250.

Going forwards I'll probably next go to a 7DmkII and the new 100-400 L, but just have to justify $4K+ to do it.

So, no matter what the rest say, I would go with the 18-135mm and see if you can upgrade the 70-300 non IS with a used IS one. You could probably get both for a little over $500 and recoup some costs by selling your existing two for around $120 or so.

In my opinion that would give you better lenses and more versatility than your current options. Then there is the new 10-18 zoom which is very sweet for the money....

Good luck.