This is an oft discussed topic on SCUBABoard.
AKR is situated on the West End of the Northern Coast, so most of their dive sites are located in that area.
The only real weather that Roatan gets comes from the North. This occurs during two seasons of the year.
1) Tropical Storm Season, August > November. In other parts of the Caribbean, this is better called Hurricane Season, but these lows almost always track to the North of the islands and smack into Belize, Cozumel and points North. Tropical Storms can shut down diving on the North side of the island with a pass as far away as 120 miles distant. AKR is one of the dive ops that has the wherewithal to transport you to the South side for continued diving. (Take the provided bus and meet the boat on the South shore~ do not ride around "the horn" in the boat of a lesser dive-op).
2) Northers are storms that appear in the Winter months, Late December, January>February. Look at the US weather map- when Arctic cold fronts push down hard on the mid-plains States, a few days later the weather mashes down to Cozumel, Belize and then Roatan. Again, Roatan's North and West are exposed. This can get real ugly.
So,
the best times for diving this exposed side of Roatan would be
March through August. I would say the same thing for the generally protected South shore, except that diving is pretty much so 365 on the South. The months of July and August can be dead flat on the North and Northwest- this zone lies in the shadow of the Island's ridgeline and is protected from breezes along the South shore. There are trade-offs to each side at that point.
If you are really planning multiple trips upcoming to Roatan, it would be a huge mis-step to only dive out of the North/Northwest. This area is good diving, but it is only typical of common Caribbean diving, certainly the absolute best of what is left of it, nowadays. To not see the South side, which is truly a totally unique environment in the Caribbean basin- that would be a huge mistake.