A modular home is primarily built off site in a factory. It is brought to the lot in pieces and assembled there. GreenPod homes are modular. They use structural insulated panels (SIPs) instead of traditional construction methods. The SIP "walls" are manufactured in a factory, then pieced together on the building site. Once it's built, you probably can't tell the difference between a modular home from a home built on site.
In the mortgage world, a modular home is treated just like a traditional stick built home.
A manufactured home is assembled in a factory. It is constructed with wheels and axles for transportation purposes. It is brought to the building site in one piece (well, two pieces for a doublewide or three pieces for a triplewide), the wheels and axles are removed, and it is place on top of a permanent foundation.
Financing manufactured homes requires special considerations which you can find here.