Q:

What is the difference between these two kinds of close votes?

We find that we can ask a post to be closed in two ways. 
Use the close link under a post.

And cast a close vote in the review panel.

What is difference between the two? How are they recorded in the count
of close votes you have cast?
One difference we can readily see is that a post appears in the review panel under close votes only after a certain process. Perhaps you would like to enlighten us on this process as well.

A:

Although lacking knowledge of internals of the SX model I have the impression, that 

clicking on the close link (let us call it `direct close vote) below the post either initiates a close review process (that is, added to the review queue at all, if it is the first close vote for that post) or just increments the number of close votes. However, this direct close vote does not count for the review queue badge (if that should be important) and at least in my case, a direct close vote of mine never appeared in my review panel, which is quite common sense, otherwise this would mean, to vote for close a second time on the same post.
The direct close vote link starts the review process. 
Voting for close in the review process increases your review count (unless skipped), regardless whether being edited/left open or closed. 

The direct close has an advantage, while being active, that it is possible to revoke one's close vote due to various reasons

The vote was just an error, being to fast with the clicking finger on the mouse
The OP edited the post, made it clearer, that it is not a duplicate, not too broad etc.

You can revoke the vote once made in the review queue (while not begin already closed) by clicking on the close button of the post.

