they have the same meaning,
"Hasta pronto" is a sentence that has no subject, it's just an interjection like "¡Adiós!"
In "Nos vemos pronto" there's a tacit subject: (Nosotros) nos vemos pronto. It's a sentence in which the verb is conjugated in the present, but to talk about a future possibility or, in this case, a wish. You can also say "¡Nos vemos!"
Te llamo cuando llegue ---> the verb "llamar" is in the present tense, but you're talking about something you'll do in the inmmediate future.
No tengo tiempo ahora, mañana te explico bien ---> "explico" is in the present tense, but to talk about the future.
I guess in both cases it would be gramatically correct to say "Te llamaré cuando llegue" or "No tengo tiempo ahora, mañana te explicaré", but they're frequently heared in the daily life :)
hope it helped!
nos vemos