In the context of asking a person to stop talking they are, as the above answers say, the same. They both mean 'stop talking'. So, I don't really agree with some of the comments that suggest you could ask someone to be quieter...being quieter is not the same as stopping talking completely! 😀 However, it is worth adding that 'Shut your mouth' can also be used to tell someone (in a very informal and direct way) to close their mouth in other contexts. For example "Shut your mouth, we don't want to see what you are eating!" 'Shut up' can't be used that way. It's a phrasal verb that means 'be quiet' - for example "I realised I was talking too much, so I decided to shut up."

There’s no difference in meaning I can think of. Yes shut up is slightly ruder, but both are and both can be used in a more teasing manner. There are lots of ways to make it more polite. Most common I can think of is Please be quiet Quiet please more polite and much more uncommon would be something like Would you please consider being quiet? A little more common maybe would be I would appreciate it a lot if you would be quiet.

There’s no difference and both are impolite although “shut up” is oftentimes used in a joking way or can mean you don’t believe someone. E.g. Person 1. I just won the lottery. Person 2. Shut up! To be polite, you could say, “ Can you please be quiet?” or “Lower your voice please” depending on the situation. If it’s an argument, you could say, “ I’m not going to have this discussion any more.”

it means the same thing and it’s both usually very rude to say. i think a politer way would be to “can you quiet down, please?” or “can you please be quieter?” however it’s generally not a polite thing to say anyways but those are some nicer was to say it.