ahdictionary.org (U.S. English) says Litigate: v. tr. To prosecute or defend (a lawsuit or legal action); pursue (a legal case). v. intr. To bring a lawsuit or defend against a lawsuit in court. So in ordinary use "lawsuit" and "litigation" are synonyms. When used by lawyers there might be a difference.

In ordinary speech, "lawsuit" is the ordinary word and "litigation" is the more formal-sounding word. If Dan G. says there is a technical difference, I believe him! I am one of those English speakers who doesn't know the difference! I think "litigation" is a noun. The verb is "to litigate." Someone who is apt to litigate is "litigious." "In the old days you could do that, but nowadays our society is so litigious that you would be worried about being sued." "Lawsuit" can be abbreviated to "suit" when it is clear from context. The verb form of "suit" is "to sue." I don't know any verb form for "lawsuit" other than "to sue in a court of law."

I think many English speakers probably don't even know the difference :) As I understand them, litigation is something you might do within a lawsuit. Or the act of bringing a lawsuit against someone. "Litigation" is a verb, and "Lawsuit" is a noun.