OK then. Just to begin up here, it is certainly subjective whether SMF or phpBB is better and they have much the same general capabilities as each other.
I personally feel that SMF is better. Here is my biased comparison:
SMF has been stronger on the security front in the past, with few security vulnerabilities found. phpBB2 had a rather poor security record, though to their credit they appear to have done much much better with phpBB3 which is supposed to be nearly a complete rewrite. They may well be comparable when it comes to security now, but I prefer to trust the people with the better track record (the SMF team in this case).
The administration panel in SMF is easier to work with while also being more versatile, though this is very much subjective. I'm sure somebody somewhere out there likes the phpBB admin panel better.
Installing and removing various mods is very simple in SMF's powerful Package Manager, which intelligently handles all of the details. For reference, the core way that it works is by a "open this file for modification, find this exact text, and then replace/add this other text to that spot" system that is able to reverse any such changes if you want to uninstall the mod, and can show you a list of all changes it will make along with pointing out problems (no code match found, or such) in the instance that any problems exist. There's no real comparison in phpBB itself; from what I recall there is a mod for phpBB which has somewhat similar core functionality, but it is a mod itself rather than a core feature of the forum and is nowhere near as advanced from what I remember of it. I have done a bit of modding of phpBB2 in the past along with a lot of SMF modding, and from that standpoint SMF is vastly preferable.
Installing minor updates/patches for the SMF software (i.e. SMF 1.1.4 -> SMF 1.1.5) itself is likewise simple. You can just go into the Package Manager and click a link to have it install the update as a mod, making changes to the source files as needed. Installing updates for phpBB (from my experience) generally requires uploading the changed files and overwriting the old ones, potentially along with any modifications you might have made to those files. This issue ties with the way mods are handled in both forum systems, with SMF updates obviously being much more mod-friendly.
The user permission system in SMF is better. Again though, this one is a bit subjective.
The template/theme system for SMF is more powerful and flexible, where the template system for phpBB is generally simpler for regular users with no knowledge of PHP to actually edit templates. So, a tradeoff there; you could indeed say that phpBB wins on that front if you're actually interested in making your own theme and just have basic HTML knowledge. On the other hand, it is generally a lot easier to install prepackaged themes in SMF than in phpBB.
The default smileys for SMF are pretty crappy, to be honest, where the default smileys for phpBB are rather decent. It's easy to install new ones though, even ones copied over from phpBB.
SMF is better designed to integrate other software, if you're interested in that sort of thing.
From my standpoint as a PHP programmer who has dug around through the source files of SMF and phpBB2, SMF's code structure seems quite a bit better, with phpBB2 being pretty kludgy. On the other hand, I haven't really checked into the source files for phpBB3. Since it was mostly rewritten, it's very likely that they have stuck to better programming standards this time around.
SMF is a little bit faster, but the difference is rather small and subjective to specific setups, mods and themes used, and so forth.
If you need help, the SMF team and community seem to be a lot more open and friendly than with phpBB. Subjective, of course.
SMF can generate RSS/ATOM newsfeeds.
SMF has a built in spell checker.
SMF has a calendar system, if you're interested in that. It can show user birthdays and "events" posted by whoever is given the privileges to do so.
If you go with the latest SMF 2.0 public beta (which is actually quite stable at this point), there is a really nice WYSIWYG posting window available.
That's my take after thinking about it a bit, but I am certainly biased towards SMF. A fan-boy if you will.
In the end though, both are good forum systems and both are free.
Again, if you don't mind help with trying to get the main stuff migrated from your current forum software, I'm freely offering my assistance. I've worked on similar projects in the past. I can also offer server space for you to run experiments on, though it would obviously be a different setup than yours and therefore things might work differently between the two.
Of course, if you want to do it yourself in order to retain complete control of the process, I'll understand.