You will inevitably get a range of opinions on this subject. What they will boil down to is the folllowing:
Packaged excursions:
The primary -- and perhaps sole -- "benefit" of a packaged
DCL excursion is the turnkey element, e.g. pay one price, get an experience where someone else takes care of all the details (transport, entry fees, reservations, etc.) In other words, no logistics to worry about.
The big DOWNSIDES of packaged excursions are (a) the "herd" element (being crammed onto boats, busses etc. and being marched around in elementary school fashion during your excursion), (b) the fact you don't use your shore time as efficiently as possible (every packaged excursion inevitably involves a lot of waiting and group transit time), and (c) cost.
Do-it-yourself:
The primary benefits are flexibility, timing and cost.
Regarding flexibilty, if you plan your own itinerary on shore, you aren't limited to the DCL "menu." As example, when we were in St. Maarten, we wanted to go to Dawn Beach. There is no DCL excursion there. Frankly, excursions are generally only to brand name, "beaten path" type attactions, and that list doesn't always inlcude the best things on every island. If you want something unique or more individual, you are almost forced into do-it-yourself.
Timing: in most cases, doing it yourself gives you more beach, diving, shopping or sightseeing time than on any DCL excursion. As example, when we were in St. Thomas, we took a cab to Red Hook, took the ferry to St. John, took a cab to the beach (Hawksnest, which by the way I strongly recommend, much less crammed than Trunk and the reefs are in better shape, too). We spent over 3 hours at the beach before returning back to St. Thomas, and still had time to shop in Charlotte Amalie. Those who took the packaged DCL excursion to St. John had less than HALF the time we did, and at busier beach.
Costs: using the same case history above, our total costs per person to go to St. John were about $37. That was less than the DCL excursion.
Downsides to do-it-yourself? Logistics. You have to plan what you want to do in advance, put together a "battle plan" to accomplish it and be willing to manage things yourself while "on the road." Understandably, that isn't everyone's cup of tea while on vacation, particularly a cruise.