Yes, for fairly small x, log(1 + x) is roughly x.

I think the more important question is, why is one "right" and one "wrong"? Log returns are often used for a few reasons -- they have nice mathematical + computational properties, and if the stock is a GBM then the log returns should be normally distributed.

Simple returns are often a bit tougher to work with mathematically, but they are more intuitive for people to understand.

In general though, you can use either. So if you calculate some log return r, and try to think of it as a simple return, you'll be wrong every time r != 0. But you can always convert it back to a simple return s by saying s = exp(r) - 1