Both generally mean the same thing.
However, since you're looking for a way to convey the message as a native speaker, I would recommend you drop both terms and instead use: I want to get some rest.
I want to take a rest
You might say I want to take a rest when you want to rest from doing something.
I've been moving furniture all day. I want to take a rest.
Hold on, I just finished a long run, I want to take a rest.
Although, in both cases, it may be more appropriate to say I want to take a break or I want to get some rest.
I want to rest
The term I want to rest is less common.
Saying I want to rest before dinner may sound better than I want to take a rest before dinner, but in either case, I want to get some rest before dinner is more likely to be the preferred choice for a native speaker.