I've been diving in Panama since 2000. My wife and I also have property and plan to live there when we retire.
The short version about diving is there are three areas to dive: Bocas del Toro and Portobelo/Isla Grande on the Caribbean and Coiba on the Pacific. When I started diving Bocas, it was better than Roatan for its variety of coral and marine life. The year-round dive sites are inside the barrier islands and shallow. You'll rarely go below 65 ft. Between September and November, the sea calms and dive ops will go out to the wall -- the same barrier reef that begins off Yucatan and ends at Colombia. The dives are deeper, and you often encounter sharks and larger animals.
Portobelo and Isla Grande are easier to reach than Bocas del Toro. I don't know if there is a dive operator in Isla Grande any more, but there are two in Portobelo. There are only four year-round dive sites in Portobelo. Like Bocas, you can get to the wall when the sea calms and that adds at least seven more sites.
The Caribbean side often has low visibility on the inner protected reefs. 40 feet is considered good. The viz is much better on the deeper sites along the barrier reef.
Both Bocas and Portobelo have been negatively affected by four things; climate change, over fishing, lionfish and tourists. The warmer water has bleached some of the coral, which drives out the reef fish. The increase of tourism -- especially in Bocas -- has put a lot more demand on fishing. Lobster, crab, grouper and snapper populations have really been hit hard. Sustainability does not interest a tourist who wants his lobster dinner -- even it is juvenile, undersized lobsters.
The lionfish hit the Caribbean side very hard beginning in 2010. Before they came, diving was like being in a fish tank. Everywhere you looked there were drums, damsels, trunkfish, porcupinefish, (lots of toadfish in Bocas), nearly everything you find anywhere in the Caribbean. The parrotfish were so plentiful that they created a racket chomping on the coral. You still see all these fish -- just not in any great numbers.
Tourism has also done a number on the coral in Bocas. Coral Cay was one of the best snorkeling spots in the world. All of it was in four feet of very placid water. There were acres covered in a tremendous variety of hard and soft corals and sponges, and with it, lots of the smaller reef fish. But, because it is so shallow, tourists stand on the coral to get in and out of the boat, or they will break off pieces to take home and they are always touching it. The water taxi drivers do nothing to stop the tourists from destroying it. So, today, Coral Cay is barely a glimmer of what it once was.
Coiba is entirely different from Bocas del Toro. Here is a report from my trip there last February:
http://www.scubaboard.com/forums/central-america/481993-scuba-coiba-trip-review.html?highlight=coiba
Some may also recommend Contadora for diving. I haven't dove there, and my Panamanian friends who are divers say, "It's not worth the trip." There are also a couple of dive ops that offer two oceans, one day and include a Gatún Lake dive. It's just a gimmick. Having seen the crocodiles that inhabit the Canal, including Gatún Lake and the Chagres River, no thanks. They take a couple of fishermen every year. Plus, the viz is barely three feet. And, the Pacific dive is nothing but a pile of rocks.
You asked about safety in Panama. There is less crime here than Honduras, Guatemala, Nicaragua or Costa Rica. Panama also has a better infrastructure, better roads and better medical facilities. There is even a Johns Hopkins hospital in Panama City.
Let me know if I can answer any other questions about Panama.
