I've not used one myself, however my father and sister have been using one in rural Georgia. Originally the problem was the house my father built apparently is really good at blocking cell signals, especially since his location was already a low number of bars anyway (The particular county is one of two in the state that has the lowest internet speeds and internet availability, so it's spotty for communication in general). So they picked up a
booster and they are able to make calls from within the house.
As for the difference, my take would be:
A
repeater would likely use a larger antenna (with a boosted signal) to connect to the cell network, then any calls your cell makes or receives is sent through that repeater rather than the cell. (A
Stingray (wikipedia.org) when in passive mode could act as a repeater, although it's also used for tracking/tapping.)
A
booster will attempt to boost the power output of a signal, where as an
extender will likely try to reduce the interference of the point to point connection of a signal. (The more interference there is the more times a packet has to be sent before a non-corrupted packet can be obtained, reducing interference reduces dropped packets thereby increasing the amount delivered over a short time period. So in some instances poor signals can actually add to the localised interference because of the attempted packet deliveries.)
the problem of course is that companies will use the terms with each other (usually to add to the word count on their sales pamphlet) For further definitions you could look at Wifi booster/extender/repeaters since they are pretty much the same thing, just instead of dealing with a cell network they work with wifi.