Well of course it's a matter of opinion - what suits one may not suit another - but as far as I am concerned, Thomson comes out on top in almost every respect. That's not to say FO are bad - they are not. One major difference is that FO do not have places to eat open round the clock. Miss breakfast by 09.30 and you'll have to wait until lunch time. And the buffets on Thonmson put those on FO to shame!
I am also booked to go on Braemar in January (first time with this cruise line). Last year we did Red Sea with Thompson Celebration, and the only fault I had was the terrible engine shudder (It was in the centre of the ship and on all levels) which went on day and night - the only time it stopped in fact was when we were berthed - for this reason only I will not go on Celebration again.
The passengers on Fred. Olsen tend to be a lot older than those found on Thomson's ships with very few children onboard even during the school holidays and therefore the entertainment is geared to a more mature crowd. Food wise I don't believe there is a great deal between the two companies although it is true that Fred. Olsen don't have a 24 hour day food option but you won't go hungry. One minor point where Thomson's score over Fred Olsen is that ice cream is not freely available on the Braemar and is only available in the evening. Bar prices, shore excursions are comparable in price on both fleets. I suspect you will enjoy your holiday. Just a word of warning USA immigration can be slow so be prepared for some delay at Miami Airport.