I am not a big fan of either location.

Bodrum is a resort town with sort of a Turkish atmosphere, but tourist development has overwhelmed the old city. Today, Bodrum is filled with British and German tourists. No beaches to speak of....the good ones are all south of Bodrum. From Bodrum, you can take a ferry to the Greek island of Kos. Kos city is beautiful, done in Moorish architecture.

In Bodrum you are within one days reach (about 2-3 hours) of the ancient Greek ruins at Ephesus, and that is well worth seeing.

Antalya is just a big city. Wonderful beaches to the east, such as Alanya, which is maybe a couple of hours away by car or bus.

If you have a couple of extra days in Turkey, I would strongly recommend that you visit Cappadocia (near the city of Goreme). It is an amazing region unlike anything you will experience elsewhere. The scenery is a surreal, almost lunar-like. And, some amazing caves to explore.