EcoQuest is right. This is the Andaman Sea's monsoon season. Some boats do operate during the monsoon based on the weeks weather forcast but 1) a lot of the best boats have moved to Indo for the season 2) boats that stay will pool customers so not always guaranteed to get the boat you booked 3) I dove the sims once in the monsoon and had a horrid experience, really rough but i know of others who did it OK. This option is open but the opertaors will only make the decssion to go a couple of days in advance.
The lower andaman (phi phi, HD, HM, Bida, etc) are open as its quite easy to head for shelter if the need arises. You can do this area as a L/board or on day trips.
John Williams (siamdivers) is a v.good recommendation, also Sunrise Divers offer good advice, booking and service.
The other side of the gulf (Koh Tao, Samui) is open during June but not great diving by any means IMHO (I live in Thailand and have not dove Tao for 4 years so that pretty much sums up my opinion).
Its a good time of year for a hol but if you are only going for diving then delay it.