After looking at both locations (and I've spent weeks in Taormina) I'd prefer Villa Carlotta. It looks like you can still walk to town from there, and I'm not so sure you could walk to town from Villa Angela. Surely you'll wat to stroll Corso Umberto and go at least to the Main piazza and perhaps see the Magnificent Greek Theatre. If not, then I'd actually recommend staying in Cefalu.
Last year we took a 'vacation from our vacation' and luxuriated in the beautiful hotel/spa Hotel Le Callette in
Cefalu, and I will tell you that our jaws absolutely dropped when, after checking in, proceeded through the hotel lobby to
our sea-side rooms -- the Views were like none I'd ever seen, not even in Taormina. They even have massages available in
your room. We never felt the need to leave the grounds, did not check out the town, we just relaxed sea-side, pool-side,
and enjoyed the music every evening and the really good meals the hotel had to offer. They changed the locations of
the serving areas for Breakfast and Dinner every day. Sometimes poolside, sometimes seaside, and sometimes indoor.
But every night at dinner, there was a pianist playing a white Grand Piano, and the food was delicious.
I will tell you that I just checked their website again, and the photos do NOT do it justice. Sorry for throwing you this curve ball, but when I realized you only wanted a view, pool and relaxation, it got me thinking! July is very HOT in Sicily.
You can easily get to Cefalu via train from Palermo or Taormina. The taxis at the train station will take you there. It is by far the best place in Cefalu. We'd seen enough of churches and museums by then!