Understanding basic operation of TM and FA
Turing machines (and RAM machines) and (most) finite automata operate in two different input models. Finite automata usually accept their input in a streaming fashion: they read one input symbol, do something, read another input symbol, do something, and so on. When the input is exhausted, they report their output. Turing machines, in contrast, get all their input on the input tape, and then do whatever processing they wish. They don't have to stop after they have scanned the entire input.

This difference in the input model explains the different semantics you have indicated. There are, however, finite automata which have input semantics similar to a Turing machine, namely two-way finite automata. Although behaving superficially like Turing machines, they are equivalent in power to deterministic finite automata (DFAs).