What is the difference between turing reductions and many-one reductions?
You got it backwards. Turing reductions are more general than many-one reduction. A many-one reduction is also a Turing reduction, but some Turing reductions are not many-one reductions. This is provably the case for (arbitrary) computable reductions. For example, there is a Turing reduction from the halting problem to its complement, but no such many-one reduction.

Completeness is defined with respect to many-one reductions since it allows making finer distinctions. Turing reductions, for example, don't allow us to distinguish the halting problem from its complement, or NP from coNP, but many-one reductions do.