Assuming you've read all of the above and still want to go, I've been to both, more than once, at all different times of year so feel qualified to advise.
Oman is less flashy and glitzy, a more beautiful landscape with natural wonders to explore. It will be much lower key than Dubai and a more authentic Arabian experience if it's something approaching actual culture you want rather than just gawping at flashy Arab money and flashy Arab sports cars. Omanis are working people and very friendly so it's easy to rub shoulders with them whereas Emiratis don't tend to do ordinary jobs if they can help it and where they do come into much contact with tourists they do they can be a bit lofty and sniffy.
The shopping and nightlife/restaurant scene won't be as good in Oman but you don't need go to the Middle East to spend your time in a shopping mall full of chain stores that are cheaper back home anyway.
Dubai is great fun for a week if you just want some winter / off peak sunshine. It's very soulless Las Vegas/Sun City type of place. All man made and modern, but impressive nonetheless.
The nicest months weather wise are October and November then March, April and May.
December tends to be sunny and bright but not warm enough to lay by the pool or go in the sea. It can rain and can be very overcast in Jan and Feb. It might be perfect weather, but it might not. Whereas if you go in my recommended months it will almost certainly be glorious. June and September are too hot. July and August don't even think about it.
March and April even if the weather is hot the sea can still be a bit chilly but if you go in October of November it will be beautifully warm as it's warmed up all summer. November less humid than October.
I'd pick November or April.