To 'acknowledge' means to recognize or have knowledge of something. You can acknowledge a person's work or contribution, for example.

To 'admit' is to take responsibility for or to accept something. You can admit making a mistake, or a country admits a foreigner to visit for a period of time.

Robert's explanation is a good one.

There are also some cases where they overlap and could both be used to mean more or less the same thing:

I acknowledge that we have a problem with spending.

I admit that we have a problem with spending.

In these cases, they both mean that the speaker accepts the following statement as true.

To acknowledge is to express recognition.

To admit is to allow or to confess.

When I entered the room, I nodded my head in order to acknowledge the guests.

I admit whenever I am wrong and ask for forgiveness.

The university admits 5,000 students per year.