I think it's a good question. With their styles and successes, it seems like they are the US/UK counterpart of each other. Both have even made songs for Disney movies (granted, the movies with John in them were better). So who do you like more? Link the evidence for your stance.

Elton john. The man can make up music to any lyrics on the spot

Elton John.

My sisters love the song Island Girl to a fault.

Billy Joel is an accomplished song writer, but I never warmed to his voice.

Billy Joel is bald so my vote goes to Elton John.

Billy Joel "She's Always A Woman" is a good song

Honestly, both have more in common than the OP described.
-Both are exceptionally good at acoustic piano (their best songs are all acoustic driven)
-Both have incredibly shitty synth tastes (just listen to: https://m.youtube.com/watch?v=Iyv905Q2omU and https://m.youtube.com/watch?v=l6Sb_QSOIVQ to see what I mean)

But Billy has always struck me as the more diverse songwriter. He can pen "rockers" (You May Be Right), can mellow it down (It's My Life), and can even put out something that doesn't even sound anything like rock (Piano Man) and it still sounds good.

On the subject of Piano Man, that beats out anything Elton has written.

Elton John's main strength seems to be writing cheesy love ballads that definitely sound dated today (Candle in the Wind, Tiny Dancer, Your Song), but he also can make a good, memorable tune (Rocket Man, Bennie and the Jets).

Granted I haven't listened to their discography apart from each's Greateat Hits, but Elton John's all sound like the same bad song over and over again. Billy slightly suffers from similar song structures like Elton does, but it's nowhere near as bad as Elton John and can easily be covered up with the right listening order.

So, Billy it is.