[susan53]

Imagine that today is Wednesday. I'm presuming that a "week" starts on Sunday and ends on Saturday.

John is arriving in a week / in a week's time / a week from now. = I expect John to arrive next Tuesday or Wednesday - ie in seven days.

He has managed to complete the whole project in a week = he started last Wednesday / He has taken seven days to complete the project

John is arriving at the end of the week. = I expect John on Friday or Saturday - ie at the end of this week

Can you let me know your answer within a week please? = You must let me know next Wednesday at the latest. If you want to do it sooner, that's fine.

Can you let me know your answer by the end of the week? = You must let me know your answer by Friday or Saturday at the latest - ie at or before the end of this week.

By and within basically have the same meaning : at or before a certain time. They set a deadline.
But notice that within is used with a period of time : within three days, within a month, within a yearand by is used with the end point of a period : by Tuesday, by March 24th, by 1987