Tiptronic is the conventional auto gearbox with the ability to shift to a manual sequential gearbox - mine has 5 fixed gears and of course reverse! Manual mode is not an immediate shift like today's DSG's.
Multitronic is based on a continually variable gear selection system using a chain and two variable diameter drive cogs, (in simple terms) - as Lostbok says they have "Viirtual" gears which forces the gear box drive cogs to a particular diameter to get a preselected gear ratio. Multitronic also offered the "manual" mode again by forcing ratios.
DAF brought this to the roads on cars back in the late 70's (DAF66 "Variomatic") where there were two rubber bands wrapped around variable diameter pulley's, DAF were taken over by Volvo who used it on the Volvo equivalents. Ford came along later with a metal chain version for the Fiesta, ..........and then Audi. On the DAF as the engineering was pretty simplistic, the gearstick was forward for forward and backwards for reverse and you could go as fast backward as you could forward!
Tiptronic is pretty bullet proof with regard to engine power, where as Multitronic was limited to a maximum engine power - not sure what this is?
An old price list I have from 2003 says Multitronic=2.0/2.0FSI/1.8T/2.4/1.9TDI(130)/2.5TDI(163), Tiptronic=3.0Q/2.5QTDI and Manual=S4