Many-one reductions vs. Turing reductions to define NPC
Two reasons: 

(1) just a matter of minimality: being NPC under many-one reductions is a formally stronger statement and if you get the stronger statement (as Karp did and as you almost always do) then why not say so?  

(2) Talking about many-one reductions gives rise to a richer, more delicate, hierarchy.  For example the distinction NP vs co-NP disappears under Turing reductions.

This is similar in spirit to why often one uses Logspace-reductions rather than polytime ones.