It's been 1.5 years since our last visit but here's a few thoughts;
For several years our standard plan was to spend christmas and new year in Anguilla for the sun and beaches and then go to St. Barth. Anguilla has better beaches and the sun is the same but St. Barth wins for everything else.
We have always rent a car because we'd be bored being in the same place for 1-2 weeks in Anguilla. Ordinary sedan is quite fine in Anguilla.
If the wind is from Atlantic side, you want to use beaches on the St. Maarten side -- our favourites are Lower Shoal Bay (CoveCastles), Cove Bay or Rendezvous Bay (CuisinArt & co). The latter is pretty good as it's very long, good beach walks, Pina Coladas on the CuisinArt beach bar and there's always plenty of room.
If the wind is from St. Maarten side, you'll want to use the beaches on the Atlantic side -- our favourite is Meads Bay although Upper Shoal Bay is also quite fine. The latter has plenty of restaurants, hotels and other tourism so it's the most touristic place on the island.
There's one exception -- the beach on Cap Juluca. It's well covered so it doesn't really matter where the wind comes from. The only drawback is that it's relatively small and there can be quite a few Cap Juluca guests. One should remember that all the beaches are public in Anguilla so you can go to ANY beach, just don't use the hotel's gear without permission.
The choice of hotel depends what kind of traveller you are, I think.
Malliouhana is a very nice property and has less resort feel. Nice sunsets and sea view.
CoveCastles has pretty quiet beach and feels more like private villa.
Cap Juluca is your typical luxury resort but relaxed, can feel a bit worn as they've had some ownership disputes.
CuisinAirt is a bit like Cap Juluca but different beach and the restaurants feel more upscale (likewise the guests, many had jewerly with them..)
For the last few years we were staying at Paradise Cove and more precisely their new building behind the swimming pool. It's a walking distance from Cove Bay but for us, it was an excellent base for sleeping. We drove to the beaches every day, sometimes switched the beach after lunch, had lunch / dinner at the various restaurants on the island or other hotels. During christmas and new year many of the other guests in this hotel are private jet staff. But I wouldn't recommend Paradise Cove for anyone who want to use a lot of hotel services or even have a hotel breakfast, it was just nice and clean accomodation for us and we bought our own fruit and bread for the breakfast (small kitchenette in the room).
For restaurants I don't rank Anguilla that high, not even the restaurants in the top hotels (we didn't try all of them, though). More expensive than St. Barth and worse in quality. The fact just is that Anguilla doesn't have much food culture unlike the french influence in St. Barth. Our favourite on the island was Trattoria Tramonto on the Lower Shoal Bay, near CoveCastles.
On our last trip we were a bit disappointed, when leaving the island we didn't feel like going back again and St. Barth felt better than ever. The locals seem to buy bigger and bigger trucks every year but don't clean their yards. The increasing tourism isn't creating lot of new shops, restaurants or similar services. The prices go up every year. A lot of new super expensive villas, hotels, condos and whatnots. Ugly buildings like Turtle's Nest Beach Resort right on the beach. "Mega projects" like the Viceroi and Temenos. What happens when the rich people find a new favourite? In St. Barth you, as a tourist, are part of the island but in Anguilla they are building even bigger gap between the tourist and the local people.