June 1 is the start of Hurricane Season for the Caribbean/ Gulf. You never know what each year will bring but historically, June is one of the "safer" hurricane months, if any such thing exist!
You will find far more to do, more anchorages and moorings, more accommodations of all types in the BVI than in USVI.
I personally like St. John in the USVI for the hiking and snorkeling and less dense sailing. St. John is about 70% National Park and lesser expensive moorings are available and most have really nice hiking trails of varying difficulty and distance. The one from Waterlemon Cay to Coral Bay is not easy but well worth the effort in my view as it the one from Lampshier Bay to the Petroglyphs. These are about a good day's trip round-trip if you stop to take advantage of the views and Danish ruins. If you don't hike then Waterlemon's snorkeling where you are almost guaranteed a great encounter with large sea turtles and a nice reef. St. James Island and it's Christmas Cove also has a nice anchorage and reef for snorkeling.
Overall, hands down most people will prefer BVI for ease of sailing, good conditions, generally less expense and a very wide range of amenities from none in a remote anchorage to large marina slips. Everything is just a short sail away if you want to hop around. About the longest hop would be from Virgin Gorda or Jost Van Dyke to Anagada which is the only non volcanic island in the BVI.
In past years when I went down I stayed several month each time and hopped back and forth from BVI to USVI but by far most of my time was in BVI. BVI is the worlds top cruising grounds for a reason.
I prefer a sailboat to a motorboat, and it is my belief that boat sailing is a finer, more difficult, and sturdier art than running a motor.... Jack London