As a beginner I find it much easier to only count when I'm playing a note. Any pros and cons?
As a beginner, we don't count at all, just play! OK, just kidding, but not really. How many times have we played something without counting knowing we should. Yeah. The great thing about this piece (shows how Bach's wife Anna or family friend Christian is a genius) is that it basically counts itself as you're playing establishing rhythm early on.
For this piece and any pieces really, you must count all the beats. You cannot count just the beats with music. Silent moments are part of the music! I don't always count, but would always still count the moments without notes. If you only count when there is music, your rhythm would be incorrect guaranteed especially when the music gets harder.
You should count 1&, 2&, 3&, 1&, 2&, 3&, out loud, even if you get tired especially when there is no music; very important to know the length of the silent air between notes.
Once you got the counting of 1&,2&,3& down, you may want to try counting loud,soft,soft, loud,soft,soft, especially when practicing the left hands alone.
The most important thing to do for this piece is to lift your hands completely at the end of each measure before starting the next measure. You want a very clean sound between measures and not have any notes held down (legato) from the end of one measure into the next.