In reply to:
For example, would a Search done in phpBB also bring up results from the WP side and vice versa? Would users just need to register in either software and they can use both?
No. The search functions in each of the programs will only search data in *that* program. As previously mentioned, you can make other arrangements for searching across both applications if that is what you desire. 
As also stated by other responders, you would need to do additional programming, or use a "bridge" of some kind to have a single user registration recognized by each of the programs; out of the box, a user would need to register and log in as a user in each program to use the "registered user" features.
--rlparker