Lighthouse Beach was across the street (a very narrow street) from our hotel. Cabbage Beach was the main drag on Paradise Island, accessible through many of the resorts.
Lighthouse Beach was calm, secluded, and frequented by many locals. When I stayed at Ocean Spray in college, we were the only people on the beach! Since then, the area directly in front of the hotel has sort of closed down . . . so with slightly less beach to use, we had some company. But still, not much. At the busiest time there were probably 15 people in the area, all at least 20 feet away. The water was calm, fairly warm, and completely clear. The view of the cruise ships was kind of cool too, since I have no experience with cruise ships.
Cabbage Beach, by contrast, was well-populated with tourists sporting wristbands from their all-inclusive resorts. The only locals here were selling something, whether it was sarongs or piña coladas. The closest towel was less than 10 feet away, so it still wasn't jam packed. The water on this side was cool and the waves were strong . . . perfect for body surfing, wave jumping, or just laughing like a child while running away.
Women traveling alone (or even with other women) will get pestered incessantly on Nassau. Everyone wants to be your "friend!" It's fairly harmless (especially during daytime at the beach), but it can be annoying. So, while the resorts are not for me, it was kind of nice not being bothered while laying out on Cabbage Beach. The waves were a lot of fun, but it was also nice being able to actually swim and float on our backs at Lighthouse Beach.
One thing: Getting to Cabbage Beach, if you're staying on New Providence, is a bit of a process. First you have to take the ferry over for $3, then you have to walk for about 10-15 minutes, past many of the hotels, going around the side of the Sheraton, through a wooded area, before you get to the beach. When you first come out it's pretty crowded, but if you're willing to walk a bit further down the beach, you should be able to find a spot to yourself.
There is no point to this . . . just pointing out differences! You'll never really know what the stretch of beach nearest to your hotel is like until you get there and experience it! Enjoy...