Can someone explain two-weapon combat to me?

The first time I won was with a paladin who was dual-wielding
Justifier and Death's Sting, and he did pretty well with them and
pretty effortlessly achieved grand mastery with the sword. Now I'm
given to understand that two-weapon combat is generally not a great
idea, but I can't understand the issue. Does it take longer (consume
more energy points) to hit with two weapons than (for instance) one
two-handed weapon, or a weapon and shield? Are there massive penalties
of some sort? What's the effect of the two-weapon combat skill?
Assuming you have two-weapon combat trained up, I assume using two
weapons means you'll hit your enemies more times within a set amount
of game time; is that correct?

Also, on the weapons skills screen, there's a skill for "two-handed
weapons". I don't think that trains if you use something like a
polearm (even though it's two-handed.) What about Vanquisher? Do you
get points for swords, or two-handed weapons, or both?