Example of something that’s different for generic and random oracles? Let $G$ be a generic oracle in the sense of Cohen / Baire category.
Let $R$ be a random oracle.

Are there complexity classes A and B with
$$\mathrm{A}^G=\mathrm{B}^G\quad\text{and}\quad\mathrm{A}^R\ne \mathrm{B}^R$$
or the other way around,
$$\mathrm{A}^G\ne\mathrm{B}^G\quad\text{and}\quad\mathrm{A}^R= \mathrm{B}^R\text{?}$$

The question was inspired by a comment by Scott Aaronson.
P = UP with a generic (assuming P = PSPACE) but they are separate relative to a random oracle.

In the other direction P = Promise-BPP relative to a random but separate relative to a generic. Can't think of a non-promise class off the top of my head.

I can track down some references if you need.

Update: If you want a non-promise version, $P^{NP} = S^p_2$ with a random oracle (because $S^p_2 \subseteq ZPP^{NP}$) but they separate with a generic oracle (example in my paper with Yamakami).