It's all about clearly stating your intent. With INNER JOIN you're literally saying "I want to join these two tables together on this particular relation and work on the result", while with the more basic WHERE form you're saying "just lump these two tables together and then we'll filter out the rows that we actually want to see". The join becomes more of a happy side-effect with that, rather than the thing you clearly want to do.

Not only does writing your code in such a way that it states your intent make it easier to read for other humans, it also makes it easier for compilers/query planners to understand what you're trying to do and turn it into a more efficient process at run-time. Now query planners are usually pretty good at distilling joins from WHERE clauses, but that form does also make it easier for mistakes to creep in that can murder your query performance in subtle and hard-to-debug ways.