If history is the reason you are travelling, Kuşadası makes the ruins of Ephesus an easy half-day outing rather than a long excursion. If you picture lazy days by the water, a choice of different bays and lively evenings, Bodrum and its peninsula are built around exactly that. Many travellers would enjoy either, and the honest answer depends on what you want a typical day to look like.
One practical point decides a lot: these towns do not share an airport. Kuşadası is reached from İzmir, Bodrum from Bodrum-Milas. Once you know where you are staying, book the flight into the matching airport and the transfer at the other end stays short.