We too have done excursion through the cruiselines and also on our own. Yes, you can save some money and have the advantage of going when you want and returning when you want. I've found that on most excursion that you book through an independant operator, its almost always a smaller group. You will find certain excursions you can only book through the cruiseline, in that case we do. There are advantages of doing ship excursions, you will get off the ship first, especially in a tender port and you know if for some reason, there is a delay, the ship will wait for you to get back. Thats not the case if you are on your own, one reason I will only book shore excursions through the cruiseline in certain ports, no savings is worth missing the boat.
In a lot of ports its very easy to hire a cab and go where you want, we've gone fron St. Thomas to St. Johns via cabs and the ferry without any difficulties, also to beaches, etc. There are always cabs waiting to take you back to the ship, you just have to allow plenty of time in order not to miss the ship. So, there are advantages to both, you just do what works for you.