what is the difference between the two Lightspeed workflows?

Please see this diagram,

Lightspeed -> BRM : to create a new reservation, and return pay for it in lightspeed.  (also Send over Lightspeed customers to BRM)

BRM -> Lightspeed : to add a new transaction to an existing reservation.  (also Create customers in Lightspeed from BRM.)

The idea is that you have two tabs open in your browser, one for Lightspeed, and one for BRM.

See also