Vim has many features that Vi does not, even features that are not obviously "advanced" features.
In practice, this means that if you are used to Vi, you will likely encounter very few differences if you start using Vim (or some other Vi clone), but if you are used to Vim and if your "reflexes" include features such as visual mode highlighting, any key action that starts with "g" or "z", any text action with "i" or "a" [e.g. "daw" to delete a word under the cursor], navigating with arrow keys in insert mode, etc, you will find that those don't work in Vi.
There's also the question of what exactly you were using when you say you "tried Vi". On many systems, "vi" actually runs Vim, in a mode where some of these differences apply (default showmode as you observed, arrow keys don't work in insert mode) and others do not (visual mode and g/z keys work), and some features depend on a compile-time option that is sometimes disabled in the "tiny Vim" that is used for this (text objects, such as "aw" a word, are one of these). You won't get these if you run the real Vi, or if "vi" is some other clone with fewer or different features than Vim, such as nvi or VILE.
And, on the obscure side, while "tiny Vim in Vi mode" obviously doesn't have any features that a full Vim does not, there are a few features of the genuine original Vi that vim lacks. These are documented in