"Who do you work for?" asks only for the name of the person's employer, and if this is what you want to know, then this is the question to ask. "Where do you work?" ostensibly asks for the location of the person's work, but will be answered based on the context, and is a far more versatile question. If I were to ask a neighbour where they work then possible replies are:
Walker's: A factory in the town - this gives both the location and the employer.
Sellafield: This is the name of the nuclear reprocessing plant. It is a huge site in which many different businesses are involved, so it does not tell me who the person is employed by.
Lillyhall: This is a nearby industrial estate where many businesses are based.
Gates's Tyres in Lillyhall: This gives the name of the employer as well.
In general, the person will probably only name the company they work for either if you are likely to have heard of it, or if it provides a more meaningful location.
However, in a different situation, the answers may be different. Asking "Where do you work?" at a trade show, for example, will probably be answered by the name of the business you work for, since that is presumably what the person is interested in (although "Who do you work for?" would be a better question to ask in this situation). On the other hand if the person asking the question is doing a survey on commuting, you are likely to answer with just the name of the town/city (Manchester, for example), or, if you commute by train, the name of the station you go to (Oxford Road).