Use the Competition Rules ("CRP"), as they're the latest ones (and also what was "Living Rulebook 6.0 (LRB6)").
They're what most of the online BB community use, so if you want help with tactics etc, that's where most of the advise is now aimed. Also if you eventually decided to run a league, you're more likely to get other BB players interested if you're using the current rules.
The rules are designed for "open" play, in that people will play pick up games, rather than scheduled ones, so there is a built-in mechanism for giving a "leg-up" for the "underdog" in any given match, which is a good thing if some will want to play it more often than others. These "inducements" aren't as cost effective as having a developed team, so there's still an advantage to growing your own.
Hope you enjoy it, here's a few handy links for you, in case you get bitten by the BB bug
: Coach's site, full of playbooks and general tactics
: TFF, formerly known as TalkBloodBowl until a GW IP purge, is (probably) the biggest online BB forum. Any help or rules questions you may have, ask here. You might also find new players in your area (and in the name of exposure, the fact I'm an admin there is not the reason I recommend it!)
: If you really get bitten by the bug, and decide to attend (or even run) tournament, then these are the guys to see. An international BB coaches society, they track and rank tourney results for coaches across the world, and run a 4-yearly international World Cup (held last year in Amserdam, with 400+ coaches from 15+ countries, from the US to NZ).
[Edit] Just realised I didn't answer one of your questions. There isn't much difference between LRB5 and CRP, mainly just minor changes for balance reasons. Most of the differences in the rosters are purely cosmetic, as some names were changed by GW (Snow Troll to Yhetee for example).