They're different models with different philosophies behind them. This is anecdotal with a user group of 1, but in my experience:

Claude has a stronger personality and is more creative. If you give it vague instructions, it's better at filling in the blanks with reasonable ideas.

GPT-5.5 is better at following instructions. If you know exactly what you want, it will do it without going off the rails. It's also less likely to imply that you're dumb, but I don't really care about that. Some people do.