1. In simple terms, the public M&A trainee will be intimate with the takeover code and interfacing with regulators (more so than their private M&A counterparts). However, you will likely be doing very similar things (e.g. research tasks, minutes, transaction management, legal diligence, drafting ancillaries, etc). The only real difference for the public M&A trainee is dealing with prospectus documents, etc.
2. If you're in a purely public M&A practice, you'll be really quiet right now and for the foreseeable future given the current economic climate. You may also be quite worried about job security. Realistically, you'll be brought on to assist / do private M&A and PE deals, etc. Otherwise, they both have typically similar hours and it really depends on the client. I would say that public M&A can be better since the clients are less sophisticated and generally look to you to guide the timing. Equally, less sophisticated clients can be really anxious and push you / your team arbitrarily hard and make your life living hell.