"Is the difference between analytical philosophy and continental philosophy one of history/tradition or methodology? both? neither?"Both. The divide that defined 20th century philosophy derives from a split in how to proceed after Kant. Very roughly, Kant argued that doing metaphysics leads to contradictions, so you have to choose between logic and metaphysics -- analytics chose logic and Continentals chose metaphysics. The continental tradition traces its roots to Hegel who abandoned the traditional two-valued logic for his dialectic. This meant two things: (1) the traditional logic no longer applied, and (2) we need to look at the objects of philosophy through historical lenses. As Continental thought reacted against Hegel and developed into the 20th century, those notions remained as the signature. The term "deconstructed" which entered our lexicon from the postmodernist school in the Continental tradition means to unpack the effects of social and political power structure in explaining why people believe something. Reality and knowledge become functions of the social.
"Why do logical proofs often use the letters P and Q? Does this have any relation to 'minding your Ps and Qs?'"Nope. Independent. The saying is British and p's and q's may or may not have been the letters p and q. The earliest usages are from the writings of playwright Thomas Dekker:
The date of the coinage of 'mind your Ps and Qs' is uncertain. There is a citation from Thomas Dekker's play, The Untrussing of the Humorous Poet, 1602, which appears to be the earliest use of the expression:In logic, we just need letters to represent properties or qualities, so we often select P and Q. Strangely, we often use F and G for relations which means logic teachers find themselves in the uncomfortable position of having to discuss the property of p-ness and the f'ing relation and whether it is symmetric and transitive.
Afinius: ...here's your cloak; I think it rains too.
Horace: Hide my shoulders in't.
Afinius: 'Troth, so thou'dst need; for now thou art in thy Pee and Kue: thou hast such a villanous broad back...
'Pee and Kue' in that citation seem to be referring to a form of clothing, but that is somewhat ambiguous. It is also not clear that the 'Pee and Kue' in Dekker's work are the same as those in 'mind one's Ps and Qs'. Dekker later used the term in West-ward Hoe, a joint work with John Webster, 1607:
At her p. and q. neither Marchantes Daughter, Aldermans Wife, young countrey Gentlewoman, nor Courtiers Mistris, can match her.
In that piece it is less apparent that 'p. and q.' refer to a form of clothing.
"We learned in your class (back when you were just the new guy on the block in the late 90's) that science can't prove anything only disprove. This begs the question (though just as a side note) if science can't prove anything to begin with than how can it disprove; what barometer is it using if proof (i.e., truth) is not counted amongst it's starting points?"The asymmetry between proof and disproof here comes from what is trying to be shown -- that some sentence is a universal law of nature, that is, something that describes the behavior of some complete class of phenomena. If the claim says that it holds for every single instance of X, and there could be an infinite number of X's, then we could never prove it. But if you show me one case where there is an X that doesn't work the way my supposedly universal theory about X's claims, then my universal claim is falsified, proven false.