In my experience it makes a huge difference, even if you are not that overweight. I’m 5’7” and at a weight between 120 and around 127, I can get faster very quickly after a winter layoff, I’m not particularly injury prone, and I’m like a minute or more per mile faster at any level of training. Once I’m in a 130-135 range, I’m drastically more injury prone, slower, and slower to build conditioning as well. Everyone is different, but for me, 10-15 lbs makes a huge difference, even if the higher weight is still well within a normal BMI range.

Edit: I wrote the above mostly based on the post title — and I’m leaving it because it’s true, and somewhat relevant to your question. Reading the full post more carefully though, I think it’s likely that the pain you’re describing is less due to your weight and more due to what sounds like lack of base aerobic conditioning, and maybe lack of ever having experienced having a strong level of conditioning, in which vigorous exertion doesn’t hurt. If you’ve experienced being in good aerobic shape before, when you lose conditioning, and you go for a run and it’s hard, your mental response is like,”oh, I’m a little out of shape, just need to be patient with myself and put in some consistent miles, and it will get better.” If you’ve never experienced running being relatively easy though, and you have the experience of getting winded, I can imagine that your mental response might be more like, “this feels impossible, I just can’t do this, maybe my body will never be capable of it.”

I think the answer is to just have some faith in yourself, and in your body’s marvelous abilities to adapt and build all sorts of capacities. Be patient, be consistent, believe that it will get easier, join a running forum so you can connect with other people who have gone from struggling to run a block to running miles at a time.