When sending a fax using a class 1 modem, the software has a lot to do with how successful you will be at completing faxes to arbitrary fax machines. The modems themselves are not equal, and the fax machines you are sending to certainly are not.
A good quality fax sender implementation will give you more successful faxes than a poor quality one.
But if you move to brooktrout - where - as I understand it - all the intelligence is on the fax board (is this right?), then assuming you can get your document converted to a TIFF and it starts to send, are you going to see the same success rates from every type of fax server software? --- In other words, is it the brooktrout board which is responsible for the success or not of a transmission, rather than the choice of fax server software?
Kind of a convoluted question, so here's the straightforward answer: Yes - it is the dedicated hardware responsible for the increase in transmission success.
Most Fax Server solutions are much-of-a-muchness. They push the data to the device, which in turn does the actual transmission work. The problem with modems is that most - some more than others - rely on software to do some of the work for them (controller and/or datapump). Better to have a dedicated device doing the single job it was designed to do.
It actually depends a lot on the Brooktrout SDK that the software vendor is using with the fax board. Brooktrout, now Dialogic, routinely resolves fax and operating system incompatibilities by updating the SDK (which includes firmware and drivers). A good fax vendor should provide these updates to you.