In a lot of dictionaries and thesaurus, these two words appear to be synonyms from different etymologies. Are there other differences in the exact meaning of the words, in its use cases, etc?
"Exclusive" means limiting in the sense of keeping out, from the Latin ex + cludere, literally to shut out. "Restrictive" means limiting in the sense of lessening freedom or scope of action, from the Latin stringere, to tie. The two words have overlapping meanings: An exclusive club is one that keeps out the riff raff, and it will also be restrictive in that it will have rules defining who's worthy to join and who's not.
But the words are used differently depending on whether the context requires shutting or binding:
Thus "exclusive" can mean fancy, expensive, or high class as in an exclusive resort, in which poor people are kept out by the high prices. In its extreme it can mean assigned to only one:
The National Enquirer has the exclusive story. No one else has the information.
But it can also mean merely separating:
The contract specifies that the company will pay all expenses exclusive of taxes.
In neither of these cases is "restrictive" a proper replacement.
"Restrictive" on the other hand is used where the limitation is important. A restrictive clause in grammar is one that defines or limits the noun that it modifies. The word is often used for regulation:
The US welfare program Medicaid has restrictive rules that dictate how much income an eligible person may receive or how much in assets an eligible person may own.
If you accept Medicaid payments, your ability to control your financial affairs is extremely limited.
In 1948, the US Supreme Court declared unconstitutional restrictive covenants that forbade property owners from selling their property to people based on the prospective buyers' race or religion.
These covenants metaphorically bound the hands of sellers to prevent them from selling their property to certain classes of buyers. This made for exclusive neighborhoods, but restrictive rules, not the other way around.