When needing to deviate due to weather, does asking for deviation "due to weather" give ATC any leeway in terms of aircraft separation?
No, it does not. ATC can specify a restriction to the deviation, especially in a non-radar environment where they can't keep an eye on you.
The only thing that frees ATC of separation responsibilities between two IFR (or IFR/VFR depending on airspace) aircraft is if one them reports the other in sight, and is issued an "maintain visual separation". After that the responsibility of not hitting the other aircraft, or its wake, rests solely with the pilot in command.