Bonjour Julie !

So...

- Il faut partir  
The obligation comes from an outside source : rule, law, time restraint...

- Il faut que tu partes     versus       Il te faut partir
In both cases, you want to emphasise who is being compelled here.
The Subjunctive option is actually the most colloquial and common here, whereas the second option is very formal and quite antiquated.

I hope that's helpful!
Bonne journée !