What do the words in bold mean? what's the difference between these two clauses?
We're GONNA NEED you.
We'll need you.
"Gonna" (used in the present progressive such as "I'm gonna" or "he's gonna") is an informal way to establish the future tense.
It is essentially equivalent to "will".
There is no difference in meaning between your two example sentences, but the one using "gonna" is more informal. You should not use "gonna" in an academic paper, for example.