Kirk had four silver bars before and after he lost command of the Enterprise and the captain of the USS Bradbury had four silver. So I presume this shows either rank or position of captain.
I would like to see an official rank table so we knew for sure.
I noticed that Noel Clarke's character had a different cap and his badge like the honour guards was plain, so I presume they were ratings/enlisted?
There was a conscious decision on the original series to downplay rank, so I think (someone will doubtless correct me if I'm wrong) that Captain's rank was cut down to 2-1/2 bars (on the sleeve) from four, and commanders were 2. Don't remember if that was abandoned or retained for TMP, though.
I imagine TheAbramsBunch weren't too concerned about this kind of thing though. Probably a little less important than determining ship size accurately.