The person recommending you is trusting you with a small part of their own reputation.

A recommendation moves risk from you to somebody else. Whoever vouched for you has told a person who trusts them that you are worth the time, and if you turn up late or handle it carelessly, some of that lands on them. It is a quiet loan, offered without terms, and rarely thought about once the introduction is made.

Treat referred work as the work you are most careful with, especially the small courtesies. Tell the person who recommended you how it went, since they are otherwise left wondering. And thank them specifically enough to show you understood what they risked.