[MichaelCollins]

I think it's a distinction that matters, morally. A regular bribe is paying an official to do something the law says the official shouldn't do. Whereas grease money is essentially paying the official to follow the law and do their job properly. The latter rewards the official for their show of tactical incompetence, but at least it isn't paying them to break the law on your behalf. More like it's paying them to stop breaking the law.

[VBprogrammer]

As far as I can make out facilitation payments are a legal exception in US law, but they are illegal in Indian law. This is definitely in the ask a lawyer before you go making this kind of payment territory.