1) The big question: Why Pathfinder? Why do you like playing it as opposed to 4E or 3.5E or another edition of D&D?
A mixture of factors: Mostly, we were driven to PF by 4E - not in a negative way: It was more a case of needing somewhere to place our pent-up new edition demand when we discovered that 4E was not our cup of tea.
Additionally, Paizo got it just right with the alpha and beta releases. I was running Rise of the Runelords under 3.5 (after our time with 4E). I checked out the beta pdf, liked the new additions and amendments (details in section 2) and found myself a print copy. I took that along to the next session and started using it as my 'core' ruleset, with a mix of rulebooks around the table.
We have tended to play 'core' only (in all editions) and thus have not experienced many of the 3.5 balance/bloat issues that others have mentioned on these boards - this meant that we were well primed for a refresh of the ruleset.
Pretty much all of the beta changes were hungrily adopted and players converted their 3.5 characters across with little hesitation. A couple more people found hard-copy betas and soon PF prevailed over 3.5.
Why might I like it to? What does it offer that other editions don't? (Besides beautiful books). What are its strengths and weaknesses?
You'll like it if you liked 3.5, basically. If 3.5 was driving you crazy, then the amends won't do it for you. If you just felt the ruleset could do with some fixes here and extra flavour/variety there than you should find something in it to enjoy. Enough to warrant a new set of books? Maybe not. But it was for us (as outlined above).
Additionally, as has been mentioned, it's an opportunity for a fresh start. So if bloat/balance issues and power-creep were problems for you, PF shakes the etch-a-sketch.
I don't think one can address the "what does it offer that other editions don't" question without straying into dangerous territory, as per Umbran's observation. Suffice it to say that it meets
our expectations of 'D&D' better than the current edition (YMMV).
Strengths and Weaknesses - I'll leave that to question 2, with an aside that for 90% of the time, the strengths and weaknesses are those of 3.5 and very much in the eye of the eye tyrant.
2) How does Pathfinder differ from 3.5? Please, no long exegesis - just short and simple with key points.
More variety in classes - there is some power creep but there are some amends that, in my view, 'fix' certain classes. Specifically: Paladins are kick-ass champions of holiness once more, clerics (through channel energy) are freed from healbotness (and the extra healing - in my experience - has led to more dynamic, rolling battles that don't get abandoned though lack of cure spells, it also plays against the 15 min adventuring day), fighters are significantly bumped. Can't speak for the new barbarian/monk/bard as no experience. I don't think it would be exaggeration to say that Sorcerors are nothing short of transformed through the bloodline rules. Rogues are given a bump with level-based 'talents'.
Some streamlining of combat, tweaks to the action economy to support new class features, the CMB/CMD is nice and straightforward. Rationalization/consolidation of skills. Removal of concentration skill. SoD/SoS spell nerfage.
There is also an indefinable 'something' about Pathfinder. None of us can put our finger on it, but we are enjoying our D&D more than ever before. Could be just damned fine game design. Could be that the host of little changes synergize to create an experience that suits us better. Could be my medication. Whatever it is, it's working for us.
The maths is still wonky though (Trailblazer territory) & things still get complex at high levels, especially for the GM (playstyle territory). I should point out that the way we play doesn't mesh with much of what I see on these boards so that's a big caveat.
Oh and... the GameMastery Guide and Advanced Players Guides are slices of fried gold.
I think that's enough one-eyed fandom for one post.