The blood clots that are being linked to the J&J vaccine are serious ones. They are more serious than the ones women sometimes get from birth control. You really cannot compare the two kinds of clotting. That being said, the apparent risk is still low. My understanding is that the reason for the pause is to go back and determine if any other serious and related complications were originally missed.
From a policy point of view, this makes perfect sense. But, from a public relations perspective it is disastrous.
The blood clots linked to J&J are:
(Even the name is scary.)
And the blood clots from birth control are typically deep vein thrombosis (uncomfortable, but not deadly).