It really depends on your priorities.
AC is by far where most tourists head. Lots of hotels, restaurants and dive shops. More shopping and nightlife there than elsewhere. You can get to Turneffe and Lighthouse from there but it's a full day trip.
Turneffe Island Resort, along with two other resorts out on the atoll, Turneffe Flats and Blackbird Caye, offers a very different experience, more like a deserted island. No cars, phones, televisions. And no shopping and no nightlife other than hanging out with the other guests. Kind of a "less is more" experience. About an hour away from the BH and LH, where they go once a week. TIR likes to brag that there are 70+ dive sites within a five minute boat ride, including The Elbow, which you might want to put on your Bucket List as well. Those three resorts are more about diving, then relaxing in a tropical island setting.
Placencia is more off the "beaten track" than AC and has the Whale Shark dives, tied to the full moon. The best months are supposed to be April and May but they'll give it a shot in March and June. Splash Dive Shop offers an overnight trip to the BH and LH, with diving along the way there and back. There are a few shops and restaurants there but not on the scale of AC. Placencia diving is on par with AC and Turneffe but not as spectacular as LH. The boat rides are longer, about an hour each way, but there are lots of small cayes to check out along the way.
You didn't mention it but if you're planning on doing any mainland stuff like jungle hiking or Mayan ruin sightseeing then Placencia is by far the best for that. It's right below Cockscomb Basin and very close to a number of Mayan ruins. And there's always Punta Gorda. You can immerse yourself in the Mayan or Garifuna culture in no time.
Depending on the length of your trip, if you're going for two weeks or more you could split it up into two of the three places pretty comfortably. There are local flights covering the entire country that make traveling quick and easy.
The good news is that there's no right or wrong answer. You'll have a fantastic time where ever you go.