Just working through it in a very rudimentary manner:
All odd numbers 1-49 are possible
Haven't found a way to arrive at 0 or an even number
Edit - Terrible explanation of said rudimentary manner:
Looking for smallest number
I took each consecutive pair (2-1, 4-3, etc.) which left 25 1s. Those became 12 0s and a single 1. The obvious end result is '1'.
Looking for largest number
The largest difference possible on the board would be 50-0=50, but there's no way I found to achieve that without another single non-zero number.
So, I took '1', '50', and each consecutive pair in between. That resulted in a 49 (50 - 1) and 24 1s, then the 49 and 12 0s, resulting in 49.
Poking around in between
From then, I just picked a couple different numbers, leaving consecutive pairs between them, which resulted in other odd numbers (1,8,pairs = 7; 1,42,pairs = 41; etc.).
Explanation for exclusion of certain numbers
I can't really explain my terrible 'methods' of trying to find a 0 or even result. I mostly just liked the question and wanted it to get some attention, then learn all the obvious things I overlooked (like @ManyPinkHats pointing out that the sum is odd, so even results are impossible).