Could I do hand layup with infusion resin?  What would be the difference other than the viscosity?

There is no difference other than viscosity, that is the only difference. But viscosity is important for hand lamination. Not only will an infusion resin drain very dramatically from vertical surfaces and pool in the low areas, but it won't fill the space between the fibres as well as a hand laminating resin meaning you will have a lot of voids in the laminate.  

Depending on what you're laminating, it might work. But I wouldn't. Infusions epoxies are more expensive than hand laminating epoxies, so it is a waste.

Thank you for the response.  I hate when I unexpectedly run out of hardener!