In-class, allowing students to address professor informally ('tutoyer' / 'tutear')

Your question is completely subjective and context dependent. It adresses the balance between formal and informal, which depends on culture, personality traits, position in the academic hierarchy, etc. It's like asking, which are the pros and cons of teaching in full business suit vs Hawaiian shirt?

Some students will feel more confident if the class environment is informal. Some won't. Some won't see the difference. You (assuming you are the Professor) may feel at ease being talked in an informal tone, or maybe not. If you are in a new country or university and need to figure out your way, attend some classes and check which is the standard. Nonetheless, whatever you chose, don't withdraw your authority as Professor.

The effects of the decision to invite your students to address you informally, and whether this is considered appropriate at all, are very culture-dependent. In general there are several trade-offs. How you balance them depends on your personal judgement. My experience is that of a German who has worked both in Germany and Austria.

Possible upsides of having the students address you in an informal manner (tutoyer/tutear/duzen) include:

• Students may be less concerned about perhaps embarrassing themselves in front of the "authority figure"; they may be more inclined to participate in class discussions and ask ("stupid") questions.
• Less perceived role conflict on your side, if you are a very junior lecturer and empathize more strongly with the student side than more senior staff.
• A more relaxed and friendly atmosphere in general

Possible downsides:

• Students may be more willing to negotiate about grades etc.; and less distance between you and the students can make it hard to say "no"
• Politely ignoring each other can be harder if you encounter students in a different social setting
• Explaining or even just giving a disappointing grade feels more personal and unpleasant when you have crossed the line between you (tu, Du) and "You" (vous, Sie).