“What’s the difference between studio and live mixers?”
In the sense that they are both just glorified signal combiners, not much. They perform the same basic function, just for different applications. Those different applications require a mixer to be able to do different things. For one obvious example, a live mixer’s inputs (however many there are) are normally mixed together and sent to a finite number of places: usually the main speakers, maybe some monitors or separate balcony speakers or something. A studio mixer, on the other hand, also mixes signals together and sends them to one or two places (a mixdown deck and studio monitors), but quite often also has a multitrack recorder connected to it, which requires its own I/O and routing flexibility.
Most of the major differences become apparent when you start to consider the workflow differences between a studio recording and a live performance, the major difference being the multitrack recorder and the need to get the proper signals in and out of it. Sometimes studio mixers have more monitoring capabilities (different sets of studio monitors, and multiple headphone mixes for talent), and a more comprehensive master section that includes enhanced talkback capabilities, a wider selection of inputs from things like tape decks, CD players, DAT’s etc.
Historically studio mixers were of higher quality and sounded better, and while this may still be true in some circumstances, both are of such generally high quality today that those lines have become blurred. Nevertheless it is often true that audio signals in a live setting aren’t under anywhere near the scrutiny they may be in a recording session. Studio mixers also were the first to have extensive automation capabilities. Nowadays, while the functionality is necessarily different for live mixers, many of them do have a suitable form of automation.
There are some things you may find on live mixers that aren’t typically on most studio mixers. For example, some modern live mixers have group matrix mixing, which make it easy for the engineer to send groups of signals (or mixes) to alternate locations in a venue. Live mixers may employ convenience features such as mute and/or VCA groups (see WFTD VCA Group and Mute Group). These make it easy for the engineer to mute and/or control the volume of specified groups of inputs through one switch or fader. This is not the same as subgroups (another feature more common on live mixers), which actually route the signal through a combined audio path. With VCA and mute groups the signals have their own routing, but are controlled by a common control fader or mute button. This function typically isn’t needed on a studio mixer with automation. Live mixers also tend to be built with portability in mind so their general construction may be much more robust.
Can a live mixer be used in a studio and vice-versa? It may not be optimal, but so long as the mixer has the inputs and outputs you need these roles can be reversed. In the past 10 years or so we’ve actually seen a large number of mixers come on the market that do a very good job in both roles.