When I was between jobs, I spent a lot of time at Starbucks. (Had to get out of the apartment!) A lot of people were interviewing there. Attacked are my thoughts based on an unofficial sampling.
Do interviewers conduct interviews differently in this type of location compared to something in an office?
Yes. It's less formal, and there's no white boarding and fewer technical questions. Questions tend to flow in both directions. It tends to be more of a 2-way screening in a neutral setting, rather than a formal grilling.
Do interviewers look for candidates to act or respond differently in this setting, and does that matter?
The interviews tend to more screens or informational.
Is it appropriate for a candidate to drink coffee or not?
Yes, just don't get an order that requires more than 3 words - it signals you as high maintenance. Avoid the "Skinny soy decaf venti chai tea latte"
Is it appropriate for a candidate to ask why they're being interviewed in a public location (like a Starbucks)?
No. But... If a candidate is very concerned about word getting out, they can ask for some place more subtle. For many candidates, it looks less like a job interview if it's at Starbucks. ("We weren't interviewing, just having coffee.") This works better at a restaurant or bar where it's harder to overhear.
Are there major advantages or disadvantages to these types of interview settings, such that the candidate should prepare differently?
Dress the same. Come with more questions, since it is more conversational.