I would say that “respite” is inherently temporary, and it is only used as a noun. “A three-day weekend is a nice respite from the rat race.” “Rest” can be both a noun and a verb: “The basis of our government rests on the rule of law.” It can be more permanent than respite: we say “rest in peace” when someone dies.

Respite is typically a break from something that is difficult to deal with or challenging example: respite from taking care of a sick relative. Rest is a break but not necessarily from something difficult. Example: resting after work.

Respite is less commonly used. It is more formal. Whatever you are going away from was probably bad. I would need respite from a sad situation. Rest is a more simple break from an activity. It would be shorter that a respite. I would like to rest after cleaning in my kitchen. I would need a respite after caring for an elderly relative for several months, who has memory problems.