Well, you've pretty much figured out the question. The answer, like so many things in backpacking, is: 'it depends.' I have a Henry Shires Tarptent - the original Rainshadow. I love it. It is light, airy, has great views, protects exceptionally well in bad weather, and will actually sleep 3 full size people and some of their gear inside. For two people, it's a palace. And it weighs all of 38 ounces total, with a floor. For hiking with my spouse and our daughter, it's fantastic.
I also have several flat tarps and bivy sacks, and love those too -- just for different reasons. For solo use, a flat tarp and a breathable bivy is a versatile combination -- use the bivy to cowboy camp or in a trail shelter, and pitch the tarp if the weather turns bad. I use this combination year-round in the Southeast.
The tarptent is somewhat less versatile, in that you either use it, or you don't. But, it's often quicker to set up than a flat tarp, and it can provide better weather protection. It's also just plain easier, especially for beginners, and it provides a large, bug-free area to change clothes, hang out, read, do camp chores, etc.
The tipi-style shelters that you mention are probably better in snow and heavy wind than either a flat tarp or a tarp tent.
You could make your own tarp-tent style shelter -- there are good plans on the Six Moon Designs web site, and thru-hiker.com sells a kit for the original Shires design.
I know this doesn't really help. I like both systems, and simply choose the best for whatever trip I am on.