I've completed Oblivion's main quest with an Imperial Knight character (Fighter's Guild, Knights of the Nine and the Arena) and Skyrim with a Nord warrior (Companions, Dawnguard, Stormcloaks, but haven't finished Dragonborn). Basically what I'm doing is playing each archetype (warrior/thief/assassin/mage) back to back on both Oblivion and Skyrim.

For those that have played all the guilds in both games, is it more fun and rewarding to do the Mage's, Thieves and Dark Brotherhood guilds in Oblivion before doing them in Skyrim? Being that the Companions were new and unique to Skyrim, there wasn't much crossover from the Fighter's Guild, but I'm betting that there is more crossover with the other guilds. (For the record, all my non-hero/warrior characters in Oblivion and Skyrim will be role-played as not being the hero/Dragonborn and will only focus on their respective guilds.)

There isn't much continuity between the two games. There are some historical references in the mages guild and dark brotherhood.

There is no Mages Guild or Fighters Guild in Skyrim so there would be no reason to expect continuity there. There are references in Skyrim's Dark Brotherhood to events that take place during the Oblivion crisis, though. If you play Oblivion's Dark Brotherhood before Skyrim's Dark Brotherhood these references will probably have more meaning for you.

Actually one of the remnants of the Mages Guild does show up in Skyrim, as a few representatives of the Synod (one of the factions the Guild split into) make an appearance in the College quest line. But that underlines that the College of Winterhold was never part of the Guild, so continuity is ruled out.

The Psijic Order opposed the Mages Guild, as they believed that high-level magic was not for the common rabble, so you can probably read something into their support, as well.