"relax" could refer to both physical and mental calming but implies more of the physical. You relax after running a marathon. "unwind" refers to mental calming only and implies that you've been very stressed mentally. You unwind after a hard day at an office job.

Hi Vicente. Both expressions are correct. The more formal is: I want to unwind.

They do have a similar meaning, but when you say 'unwind' we understand that you want to relax AFTER something - usually a long day at work. You come back from work stressed and 'wound up', so you have a cool drink and go for a swim in order to unwind. The meaning of 'relax' and 'unwind' are not always the same. 'Relax' suggests putting your feet up and doing very little, whereas 'unwind' can be very active. For example, a ten-kilometre run can be a way of unwinding after a day at work. This can help you unwind, even though it's not necessarily relaxing. 'Unwind' is a kind of image/idiom, so it is more informal than the neutral term 'relax'.

Yes, they both are correct. I want to relax may be a little more formal, but basically they are the same.