For the end result, no. Software can make a difference in how fast you get to the result you want, though. For example, animating in MS Paint is a horrible waste of time.
Some programs also make certain kinds of workflows and types of art easier than others. For example, some programs let you preview your tiles as you make them, so if you do a lot of tile work, you'll want one like that.

So yes, software matters, but the software that you should use depends on your working style and what you want to make. There is no "best" software.

I currently work in Pixen. It's a macOS editor. For the most part it's great but it lacks some features like limiting color palettes. So this is an example of something that could make my work faster but in the end I can allways count my colors by myself;)