In which context these words find their use? are they synonym to each other?
Rest of the things is used when we do not know the quantity of things.
On the other hand, Remaining is used when the quantity of things is known to us.
Let's clear it up with a couple of examples.
Lets assume that in case one I do not know how many pages a book contain.
1) I've read twenty pages today, I'll read the rest of the pages tomorrow.
And in case two we know that a book contains fifty pages.
2) I've read twenty pages today, I'll read the remaining thirty pages tomorrow.
Above was just an example, you can substitute the pages with whatever you wish.
I would say : "Rest of the things" is used when you talk about many things but are not supposed to do some action with the rest of the things. "Remaining things" will be used in a situation where suppos you have got 5 tasks to do and you have completed 2 , 3 are yet to be taken care of. So you are supposed to perform some action on the remaining 3 things/tasks.
They are not very significantly different though.