Galle and Unawatuna are five minutes apart by tuk-tuk, which makes the choice between them easier than it sounds. You are not committing to a location that cuts you off from the other. But they are genuinely different in character and the right choice depends on what kind of trip you want.
Stay in Galle if you are here for culture, food, and atmosphere. The fort is the defining feature: a walled colonial town with Dutch architecture, independent restaurants and cafes, a working lighthouse, and ramparts with views over the ocean. The evenings inside the fort are atmospheric in a way that Unawatuna's beach strip is not. Galle has the best restaurant concentration on the south coast in a small area, and the variety of accommodation inside and around the fort is strong across price points. If you want to feel like you are in a place with history and character, stay in Galle.
Stay in Unawatuna if you are here for the beach. Unawatuna's bay is wide, sheltered, and one of the best swimming beaches between Galle and Weligama. The beachfront has a relaxed, slightly party-oriented atmosphere with beach bars and dive operators. Jungle Beach is a ten-minute forest walk from the main strip and significantly quieter. The Japanese Peace Pagoda on Rumassala Hill is a short tuk-tuk ride away. Unawatuna is a more conventional beach destination than Galle.
What Unawatuna does not have is the character of the fort. The accommodation is generally cheaper than inside Galle Fort but the surroundings are less distinctive. The main beach strip in Unawatuna gets noisy in the evening and the restaurants are aimed at the tourist market in a way that Galle's better spots are not.
What Galle does not have is a beach you can swim from easily. You need a tuk-tuk to reach Unawatuna or Jungle Beach. If swimming every morning from your door is the priority, Galle Fort is the wrong base.
The practical answer for most travellers: stay in Galle for culture and food, stay in Unawatuna for the beach, or split two nights between them. Both are easy day trips from each other. The first night in Galle Fort followed by a move to Weligama the next day, with Unawatuna visited as an afternoon stop in between, is a route that works well.