What's the differences between the two?
Personally, I'd say let's meet each other at three o'clock. Is formality the only difference between them?
Here is the complete conversation:
I don't think formality comes into it, but these two individuals seem to be under the misapprehension that it sounds more formal.
You could argue that, "planning to meet," allows for some flexibility, and the understanding that the plan may have to change, but in reality that would also be the assumption for, "let's meet at three," as if something comes up, plans may need to change.
I would put it down to the facet of human nature that likes a little bit of self importance. "Planning to meet," is just a bit redundant.
To me the difference between your two sentences is merely style. They mean the same thing. The one with "plan" is verbose: "plan to" is unnecessary. But perhaps other native speakers will say that it adds not so much formality as definiteness to the meeting at three o'clock. Making plans seems to be more definite and less spontaneous than simply "Let's meet at three..." That sounds reasonable to me. I wouldn't disagree.
The first statement - "lets plan to meet at three o'clock" - is hedged; the second - "lets meet at three o'clock - isn't.
What this means in real life is that the first statement is less definite and less assertive, and possibly leaves a way out if the speaker suspects he may not be able to make it. It is similar to the difference between close the window and do you mind closing the window.