There are two main differences between Do Not Disturb and Bedtime mode. First, Do Not Disturb will silence all incoming calls, texts, and notifications while Bedtime mode will only silence notifications. Second, Do Not Disturb can be set to automatically turn on and off at specific times while Bedtime mode must be manually turned on and off.
Do Not Disturb is a great option if you want to completely disconnect from your phone and the outside world for a period of time. It's also useful if you have specific times when you don't want to be disturbed by your phone. For example, you could set Do Not Disturb to turn on from 10pm to 8am so that you won't be bothered by any late-night calls or texts.
Bedtime mode is a good option if you still want to receive calls and texts from certain people (e.g., your family or close friends) but don't want to be disturbed by other notifications. With Bedtime mode, you can also specify how long you want it to last (e.g., 30 minutes, 1 hour, 2 hours) which can be helpful if you know you'll only need a short break from your phone before going back to sleep.