OK, SMF is the best choice in free forum software currently.
The only viable alternative I know is Fireboard for Joomla - a CMS plugin.
Let me explain: all forums I've tried are restricted, old-fashioned, have major owner-usability issues, poor backend admin, lousy SEO with or without plugins, no on-page assets at all, etc etc, compared with modern CMS and blog apps. They are an order of magnitude worse in every respect. The least-bad option is SMF.
vBulletin is an exception but it's commercial, you have to pay for it and also many of the plugins. So if you have the budget then go with vBulletin. It's dirt cheap as far as an enterprise is concerned, of course, so basically, any business should use vBulletin.
Non-profits will do OK with SMF. It's the best of the free solutions. However I have to qualify that: I'm not a forum software specialist and I haven't tried them all. For example I haven't tried that one you mention your host provides: Burning Bord Lite.
But personally I would be wary of anything with little history / background as the possibility for exploits in a forum is about the highest of any webapp.
The only exception to the 'free forum apps are stone age in most departments' is Fireboard. It's a bit crazy to recommend a CMS when you need a forum - but because Fireboard is so much better in every respect than the purely forum-only apps, I don't have a problem with that. It's better because the CMS that contains it is light years ahead of the forum apps in every possible department. There's just no comparison, it's like comparing a horse and cart to Concord or something. A different world. The only negative I can think of with Fireboard is that the ACL is no good (IMO), it's a good solution if all your boards are open to one user level and there are no private boards.
You could probably install Joomla + Fireboard and have no CMS content if you wanted - just the forum. But I don't think it would stay like that for long.
It's why I install a CMS on the front end of ecommerce sites - you need that stuff and nothing does it like a good CMS can.
I don't want to talk about phpBB as I said because there is nothing positive there. The exploits situation is the most worrying part. See my review on the main site. SMF has no history of exploits as far as I know.* It's not only rare for Google to go as far as naming bad web applications, it's pretty much unique as far as I know. So if G tell you that something is bad news, you could take that as being a bit more than a broad hint.
* Well, of course every webapp or anything else that touches the Net has been exploited, but the point is: how often, and what did they do about it, and how good is it now.
In answer to your question, exactly why is SMF best, the answer is basically, it's the least-worst option. That of course is the opinion of someone who has been exposed to the ultimate in modern wepbapps, the best of the open-source and commercial CMS projects. It's very hard to touch the best open-source ones as far as quality issues go, there are few commercial CMS that even approach those quality levels - with a few exceptions. A lot of commercial CMS is junk, quality-wise. There are too many developers still living in the stone age, they don't have a clue about quality or commercial reality.
But I shouldn't complain about that, of course, as 50% of my work is getting paid to fix poor development. So keep it up guys
