Each of the islands is a proudly independent nation. The idea of, say, Jamaica teaming up with Barbados and Antigua for sporting events is a bit like expecting the UK to team up with France and Germany to enter a combined team in the Olympics.
However, the individual islands don't have large enough populations to field a cricket team at Test match level. So they field a joint team to enable them to do so. Even the name is inaccurate though, because Bermuda (which contributes players to the team) is nowhere the West Indies!
Things aren't so different here either (except that they seem to be reversed). For example we field separate cricket teams for England, Scotland and Northern Ireland, whereas the 'England' team includes Wales. At the same time we have a United Kingdom team competing in many sporting events. In the Olympics we field a team called 'Great Britain' but include athletes from Northern Ireland (which isn't in Great Britain), even though those same athletes can compete for Ireland if they choose to do so. Nothing's ever simple, is it? ;-)
Chris