Both “super” and “superb” are common, but they are not completely interchangeable. Super is used nowadays mostly as an adverb meaning “very, really”, for example:
She was super friendly. = She was very friendly.
Nevertheless, such usage is considered colloquial and should be avoided in formal writing. As an adjective, “super” is an informal way of saying “extremely good”:
However, “super” in this sense is falling out of use and may be considered old-fashioned, so, if you are a non-native speaker, you can safely avoid it.
Superb, on the other hand, means “excellent, outstanding”. It is not informal at all; on the contrary, it is mostly used in formal contexts:
The singer amazed us with a superb performance. = The singer amazed us with an outstanding performance.
Also note the difference in pronunciation (“super” is stressed on the first syllable, “superb” on the second):
superb — US: /suˈpɝːb/ or /səˈpɝːb/; UK: /suːˈpɜːb/ or /sjuːˈpɜːb/