In the graphics cards that I have worked with, I have seen hardware 3d acceleration in all of them, and even on the integrated cards. Though I dont know about ALL of the modern-day cards, I would assume they do, but that is only an assumption.
There is a pretty big difference between shared memory and dedicated. Obviously dedicated is better. Its like a server, its designed to server and only server clients. Then if you start using it for home use, like surfing the web just because or downloading files and playing games, movies editing..... it becomes less efficient at server, because resources are being shared.
I dont know if that is either to simple of an example for you and you feel as if I am treating you like a little kid or that is way beyond you.
Say you have 128MB of RAM in your computer and you need 8 of it for you integrated video card. That automatically slows your system down, because it delays the processing of information. If this happens is takes longer for your computer to figure out what it is doing.
But now you have a Graphics card, there is a GPU on it, which helps with some of the stress on your CPU and then it has it down dedicated memory. Obviously this is better.
You can think of the difference between the two like this:
1. Computer with Integrated Gfx has a 4 lane high, 2 lanes for actual memory use by the CPU and then 2 lanes for the gfx
2. Computer with Ggfx card has TWO FOUR lane highways, 4 lanes for memory usage for the computer, and 4 lanes for the GPU.
I dont know if this helps any, I tried. If I missed something or screwed something up let me know.