I am referring to bi-weekly payment. When we refinanced our TH,
we got a flurry of calls and such from our mortgage holder trying to
get us to do biweekly payments of half of our monthly payment, saying
that we'd pay off the loan early.

This is true since you are actually paying more with biweekly payment
(there are 4.3 weeks in a month), but since the loan is a 30 year
fixed, we figured that we wouldn't save anything since we don't plan
to live here more than a few years before moving up (we live in a 2
bedroom TH), and were wondering what was in it for the bank?