Fighter 1 / Barb 1 / Fighter X probably kicks butt on both straight Fighters and straight Barbarians.
He gets the rage for the big fights (like Dragons), has better AC in the long term, and fighter benefits on average just seem better than Barbarians (e.g. 3 attacks per round at level 12 for the multiclass without the exhaustion of Frenzy).
Keep in mind, a lot of that early Barbarian stuff, such as Rage, Fast movement, does restrict on heavy armor.
That said, having Action Surge as a fighter is really strong early levels - recharges on short rest. Both have their pluses and minuses. Rage on Long rests...
Grabbing 1 Barbarian as first level, then Fighter is what I am doing for the ultimate tank, 4 Barb Wolf Totem, 16 Battlemaster. Giving all allies advantage on whatever is within 5 feet of you, having physical damage resistance, having advantage on DEX STs (coupled with resilient and a Dex build). It adds up quick. Going unarmored, grabbing defensive duelist, protection from Fighter. With Maneuvers to throw around to boot.
Hands down, best tank in the game. I just rolled one for Adventurers League, 27 point buy, 14 STR, 15 DEX(+1 Human), 16 CON (+1 human), 8 INT, 8 WIS, 12 CHA and starting with the healer feat. It is going to be a really fun character.
I do feel multiclassing between the two is going to be common if you want to give up heavy armor, if you don't then it becomes not really worth it. IF you want the heavy, stick with fighter, you just are not gaining from barbarian otherwise.