Is one more polite than the other? Is there some situations where you'd use one or the other? Would different people prefer one?
In my generation, トイレ is definitely the most common word refers to that facility, over any native word. お手洗い is also usually heard, but whoever says お手洗い in daily conversations would be judged being overly polite or really well-born. Nonetheless, we hear お手洗い more often than not because it's very popular in salesperson-ese, thus you're quite likely to run into such a situation:
As other people already said, there are two more names of toilet room you may encounter in Japan today, but both aren't really spoken language.
便所: This word makes me feel either outdated or vulgar if spoken. Maybe because a typical 便所 I see is shabby, hoary, undermaintenanced equipments like the most of 公衆便所 standing outdoors.
化粧室: Most graceful word that only seen on signboards. Literally meaning "powder room", it might be a bit funny if you look at apparent characters 男性化粧室, but not many people seem to care about it.
For public toilets, I remember that the toilet signs at the 明治神宮 and several other 神社 say お手洗い, not トイレ. I think you'd have a hard time finding a toilet in a 神社 called トイレ. Similarly そば屋 usually use お手洗い. But I wouldn't be surprised to see a toilet in a そば屋 (especially in a cheap one) called トイレ.
On the other hand, most デパート I've been to use トイレ, the same goes for railway toilets (e.g. of 東京メトロ or JR), but I wouldn't be surprised to see お手洗い there.
There are two other words for toilet: 便所【べんじょ】 and 厠【かわや】. 便所 is maybe like "loo"; お手洗い is probably somewhat more polite. 厠 isn't used much anymore and I would be surprised to see it anywhere but on outside toilets.
御手洗【みたらい】 is the purification fountain at the entrance of shrines, where you purify your hands and mouth before entering.
御手洗【みたらい】 is also a family name.
The area for washing hands is called 洗面所.
トイレ is a shortened form of English “toilet” in Japanese. Both ‘お手洗い’ and ‘便所’ are Japanese language equivalents to toilet or lavatory. I think ‘トイレ’ is becoming more popular than ‘お手洗い’ and ‘便所’ these days. I think older generation, particularly females tend to use ‘お手洗い’ in the way ‘ちょっとお手洗いに行ってきます.’
便所 is almost obsolete today. No one use 厠 – kawaya (equivalent to Chinese 厠所 which is current in China) today, though it had been used before the World War II, and appears in literature.
化粧室 is also used as an Japanese equivalent to powder room. But it sounds somewhat standoffish. I’ve never heard someone says ‘ちょっと化粧室に行ってきます.’
The difference between those 2 words could also be akin to the US-style 'washroom' (euphemism for toilet) and the direct term 'toilet'. For me, お手洗い could be broken down into to お手 (hands) & 洗い (wash) i.e. 'wash hands' in the washroom, i.e. euphemistically visit the toilet.