Yes a new mattress can help you sleep better especially if your current old mattress is 8 years old to 10 years old or older.
I always get a new mattress after 8 to 10 years and when I do I find I sleep much better and wake up feeling more rested.
I also don't have as bad back pain when sleeping on a new more comfortable mattress.
I prefer the memory foam mattresses though over the spring mattresses as the springs in the mattress can get hard to sleep on.
But after switching to a memory foam mattress I sleep much better and it's more comfortable.