I've been visiting Panama for various lengths of time for over 20 years, and I've been diving there for the last 15. Last year, I finally dove Coiba for the first time, and I'm looking forward to going back.
Coiba is an adventure. What you'll see there has very little in common with Bocas del Toro and the Caribbean side. Coiba has deep dives, thermoclines and lots of large animals. Bocas (in May) has shallow dives (max 20 meters), bathtub warm water and small animals. Off Coiba, you'll see lots of reef sharks, large schools of barracuda and jacks, turtles, a large variety of eels, frog fish and coral in small patches. In Bocas, you'll see the smaller Caribbean reef fish, a large variety of coral, and you'll be very lucky if you see a nurse shark or a turtle.
Bocas was once a backpacker's haven (it still is), but over the last 10 years -- and especially the last 5 -- it has seen a surge in development with lots of condos and "natural" resorts that have turned it into much more of a tourist location. Santa Catalina -- the gateway to Coiba -- is still a small, undeveloped haven for surfers and divers and some who never left. It doesn't even have a gas station, so make sure you fill your tank in Santiago or Soná while on your way to Santa Catalina.
If you do choose Coiba, stay on the island. There are limited amenities (you'll sleep in an open barracks room with your fellow divers, and the electricity only available between 7:00 PM and 7:00 AM), but just the wildlife alone makes the stay worthwhile.
You also need to consider how you're going to get to either location. There are three ways of getting to Bocas del Toro. One, take an hour-and-15 minute flight on Air Panama from Panama City. Round trip tickets are over $200. Two, drive a rental car to Almirante where you'll park it and take a water taxi to Bocas Town. The drive from Panama City to Almirante takes over 10 long hours, and you have to get to Almirante before the last water taxi leaves at 6:30 PM. (Almirante is not a place you would want to stay overnight.) The cost of the rental car will probably make the Air Panama tickets look like a bargain. Three, you can take the bus. There is a direct, overnight bus from Panama City for something like $10 - $15.
Getting to Coiba (Santa Catalina, actually) is also an all day event. It's a 7 hour drive from Panama City. Or, you can take a series of busses. You catch the last one in Soná, and it only goes to Santa Catalina two or three times a day. Even though the bus is far cheaper, I recommend driving.
I also posted a trip report
here.