I will stay up late to watch/watching a movie.

[Tait-ka]

Me to a friend: Tonight I will stay up late to watch/watching a movie.

I created the above sentence.

Can I please ask the difference in meaning?

 

[emsr2d2]

Me to a friend:

1. Tonight I will stay up late to watch/watching a movie.
2. Tonight I will stay up late watching a movie.

I created the above sentences above.

Can I please ask the difference in meaning?

Note my changes above. I crossed out "Me to a friend" partly because it was unnecessary but mainly because no would say the following words to a friend: "Tonight I will stay up late to watch slash watching a movie". That's what you suggested happened!
Don't make questions more complicated than they need to be. Your final question would be much more natural as "What's the difference in meaning [between them]?

1. Using "to watch" explains the reason that you plan to stay up late. You're going to stay up late and then you're going to watch a movie.

2. Using "watching" expresses what you'll be doing at the same time as staying up late.