The only thing a 788 has in common with a 700 is both made by Remington, and all the 788 chamberings are also found in the 700.
The 788 action uses multiple, small locking lugs at the REAR of the bolt. The 700 uses a pair of large locking lugs (like a Mauser) at the FRONT of the bolt.
There are other differences but the main one is the actions are simply different. ALL 788s use a detachable box magazine. Detachable magazine 700s are very rare, most have a fixed magazine, either a blind box (no floorplate visible on the stock) or have a hinged magazine floorplate.
The 788 was designed as a budget rifle, the 700 is the "flagship" model.
788s have a reputation for being good shooters, but to me they are rather ugly looking.
Certain caliber 788 magazines are not easy to find, and can be rather expensive.