Each are used almost interchangeably, but there are differences. Not all highways, for example, are interstates, nor are all Freeways.
Turnpike is an old phrase to denote a toll must be paid. The toll collector/guard would approach the traveler, his like at the ready, to collect the fee. Once paid, he'd turn his Pike aside and let the traveler continue. This lead to non-toll roads to be referred to as the pike, to differentiate it from the toll road, or turnpike.
The phrase is used mostly in the Eastern States.
You don't want to get me started on parkways and driveways