Yes a palmetto bug is the same thing as a cockroach.
The palmetto bug is a species of cockroach and is the American Cockroach so if you're in America and have cockroaches you likely have the Palmetto bug version and species of cockroach.
The palmetto bug or American Cockroach is a large winged cockroach and prefers to live in damp and dirty conditions and can often be found in sewers, woodpiles and mulch.
You should make sure to keep food off the counter, floors, stoves etc and spray for cockroaches as soon as you see them.
I would recommend hiring an exterminator if you start to get cockroaches because cockroaches once you get them can be hard to get rid of.
Even if you have a nice clean house you can still get cockroaches.