I like on national fan pages that you can see pics from various chapters, etc.
I also like the idea of fan pages as a promotional tool.

For anything private, keep it on a locked group or off the web entirely.

I'd say don't ever say anything the least bit private on Facebook.

My chapter does both. The fan page is used as a promotional tool / flyer, whereas the group is used for identity and lightweight communication. Basically, the fan page is presented to outsiders, whereas the brothers (and pledges) use the group to say that we're all in the same fraternity. There's very little actual use of the group, but we do pass around a few photos on it.

As for anything else, like planning, things relating to individual members, etc., that's what chapter meetings are for.