Here is a feature comparison between the two. In my opinion, phpBB3 is better in every way.
And yes, you can upgrade from 2 to 3 without losing any posts or users.
If you have not installed either yet, it would not be wise to install phpBB2. It reached end of life last month. It is no longer supported by the community here or the developers. There will be no more security updates for it, etc.
I'm looking to upgrade a board from php2 to 3, but I wanted to test this in a safe area first so I know what to expect. Is there any place I can download php2, so that I can run an upgrade?
Not a problem, already downloaded and install phpBB3, so I'm going to scrub the install and mysql tables, then start again so I can do a full test and re-craft the existing templates to fit v3
I see this has not been mentioned here, the phpBB2 branch has reached it's EOL and is no longer supported, even the date for releasing of security patches has passed.
I really recommend using phpBB3, not talking about the features and possibilites it brings, it can handle a higher load etc...