Difference between "plenty of money/friends" and "a lot of money/friends"?

Solution 1:

"Plenty" is more about relative quantity. Regardless of the absolute quantity of friends/money/time/whatever, it's enough for what you need. Whereas "a lot" is a statement about a large absolute quantity without a judgment on the value of that quantity.

E.g.:

Me: I have fifteen dollars to spend on this dinner.
You: Oh, that's plenty. (meaning: more than enough money for what you intend to do with that money)

vs:

Me: I have five hundred dollars to spend on this trip.
You: Oh, that's a lot of money. (meaning: that's a large amount of money in absolute terms, but I'm making no statement as to whether it'll be enough for your needs)

Solution 2:

In the first sentence, the world plenty indicates that the amount of money/friends the speaker has is sufficient for some purpose, but doesn't specify whether that amount is generally a large number or not.

In the second sentence, the phrase a lot indicates that the number is generally high, but not whether it might be enough to meet some goal.

For example:

I have plenty of money for a cab, but I don't have a lot of money.