Ĉe is "at" or "near". It can also mean "in the home of".
Apud is "next to".
Consider the phrase:
- Mi sidas apud vi ĉe la tablo.
I would not recommend using "apud" for "at the table". I found a few examples of that, but it strikes me as odd - possibly archaic. Sometimes it's clear that the sense is "next to the table" ("standing next to the table") and other times it does seem that the author was trying to say "ĉe".
PIV clarifies that the difference is the same as between "super" and "sur" - which to me means that "ĉe" is closer -- as if touching.