I'm pretty sure the "P" stands for "proportional" (i.e. non-monospace) characters; other than that they're identical to their non-P counterparts.
"UI" is the same font tweaked for appearance as a "user interface" font (appearing in low font sizes as window titles etc). Seems to be proportional as well.
Personally I prefer Japanese characters to be monospaced. I avoid the proportional ones completely.
There's a noticable difference between the Japanese characters in MS Mincho and MS Gothic at font size 12 - Mincho has serifs and is more "curvy" on the kana. (Look at あ at font size 12. In MS Gothic the horizontal bar is flat on the left, curves up on the right, and is much longer to the right - but in MS Mincho it curves up at both ends and is only slightly nonsymmetrical around the vertical bar. And the vertical bar in MS Gothic is completely vertical, but in MS Mincho it curves right at the end.)
Also, MS Mincho and MS Gothic seem to have no antialiasing/smoothing before font size 16, whereas Meiryo always has it.
Meiryo's Latin characters are basically Tahoma, unlike the other two fonts. I prefer Meiryo's Japanese characters to MS Gothic, as well. Meiryo scales better (retains its form across sizes), whereas MS Gothic has "jumps" in its appearance between font sizes 8-9, 11-12, and 16-17. (After 16 MS Gothic seems to suddenly change weight XD) And the kana of Meiryo just look nicer to me.
They're all mainly body fonts so it's not surprising that they're all fairly simple, thin in weight, and reasonably similar in appearance, but there are still definite noticeable differences.
The question of "which do people most use when typing" ... you can't tell online

Since you can't normally pick the font. Unless you're writing a blog and customizing its appearance. So I'm not really sure what opinion is there.
That's a pretty limited selection, though, and there are many other (free, even) Japanese fonts out there. It takes a little bit of searching to find ones containing a full (enough) range of kanji. But I think there have been threads on that... ._.