First, would you say there is a great difference between the term "core" as used by someone like say, Joseph Pilates, and the term "Dantien"? What would you say the major differences or similarities are?
Second, what do you think is the major advantage of using a muscular structure (Dantien) located in the abdomen, over using localized muscles? For example the muscles of the legs when doing leg presses over the Dantien. Or do you simply believe that having a strong musculature in the core area of the body will help the functioning of all other muscle groups as a whole? This would be using the core muscles as a stabilizing influence on the body as a whole.
Yes, there is a difference. Depending on what you are referring to as a dantien, I believe there are up to 7 dantiens, some of which correspond to chakra points like in kundalini yoga. From a martial perspective in terms of dantien usage, I'm not sure if all of these are applicable to the discussion.
I believe your question is centered more around the lower dantien. Now while pilates and internal strength both work the psoas, the psoas have more to do with the idea of the kua/dang. Thats not to negate their importance or usage, but think of the dantien as a control center in addition to typical muscle usage. Now from what I have been told, once you have control over it, the logic becomes clear and you really can move the arms with the dantien. I personally can only use mine to open and close the hips and tug on my arms a bit. Thats likely because the front side of my body is weak and my backside is strong (not shearing forces), so I will have to work on that more before i can get more access over it.
The lower dantien in this case is a kinda golfball sized area below the belly button. It is not the entirely of the abs, psoas, and pelvic floor. While I have never done pilates, I believe this is what is generally referred to as the core.
To answer your second question, think of it as a control point. That is movement in the entire body originates in that point. Pushing or pulling with this area is like squeezing a balloon animal, use it to apply pressure to stretch/comrpess one area, and it inflates another. With respect to the leg press, clearly the legs have the ability to take more weight than the dantien due to their size. You have several advantages with the dantien, you can push first with the dantien which causes the "suit" (think the balloon description above) to move and take up the load. Pushing with the dantien also causes the hip area to push as it forces them to open up then the legs etc, so as a result you have recruited more muscle.At least for me, if I try and do it with the abs, I can't get my abs to force my psoas to exert force; that's to say I have to work them separately.
Clearly having strong abs adds to stability, I think thats needed for either approach.
There are people here who have focused more on dantien centric movement than I have, hopefully they can chime in some.