So, what is the difference between a Fighter and a "Warlord"?
I've never played or even seen the class, so I am honestly curious...
Well, it was only around for two years, so it was easy to miss.
There really
shouldn't (if I can get all design-philosophical for a moment) be a difference between the Fighter and Warlord. Nor the Fighter and Rogue, or non-spell-casting-Ranger/Scout (if you're familiar), or Monk, or various 'Swashbuckler'/'Duelist' or Bandit or other non-casting classes that have popped up over the years. They're all just variations on the archetypal Hero of myth/legend/literature and the broader fantasy genre - and, unlike supernatural classes, even history.
But, very early on, the Fighter was narrowed to a mindless DPR-grinding wall of muscle, with zero support for much of anything beyond hitting things with a stick (OK, a magical longsword). Whether that resulted from dungeon-exploration skills being initially granted as niche-protected special abilities to the Thief and snowballed from there, or whether the Thief was just the result of that conceptual narrowing, is a chicken-egg thing, I guess - personally, I blame the Thief.
Either way, the upshot has been that, throughout D&D's history, supernatural options have generally been expanded by adding to existing classes' spell lists as well as adding new classes & magic systems, while martial options have generally been expanded only by adding new, often very one-trick, classes. The Warlord was one of those. Rather than giving the Fighter mechanical support for the nominal role of 'natural party leader' it was fluffed as having in 3e (let alone the budding-feudal lord concept it had in 1e and basic &c) and that was common to so many of the archetypal examples that the fighter was meant to represent (especially those drawn from history, who were often generals & the like), that support was given to a new class, the Warlord.
(It was also a consequence of 4e dividing classes into formal 'Roles.' The Warlord was the Martial version of the traditional band-aid cleric, more broadly, a support role that synergized with the rest of the party, it did so, in the PH, via either Tactical coordination or Inspiration. Because 'healer/buffer' and 'cleric' and 'support' had accumulated some bad press and dissatisfaction over the years, the role was re-named 'Leader,' though, it was made abundantly clear, up-front, that it did not mean bossing the party around.)
In 5e, the fighter is as limited to straightforward single-target DPR as ever (moreso than in 3e or 4e - really, it's strongly reminiscent in it's basic contribution to the party, to the late-1e and 2e weapon-specialized fighters, minus the expectation of collecting a lot of magic items to round it out and help it keep up with monster special abilities). So 5e still looks to additional classes - Rogue, Barbarian, Ranger, Monk, Paladin - for any variations on the archetypal Hero, all too often, adding obligatory magic or spellcasting to the mix.
A warlord is a cleric who casts spells without using magic.
A prejudicial misrepresentation often repeated in the edition war.
The fact is, no Martial class cast spells. Period.
No martial class used implements (spellcasting focus in 5e). Period.
No martial class had an attack or utility that was subject to Dispel Magic. Period.
(Technically, in fact, only Arcane classes actually cast "spells," the supernatural powers of the Divine, Primal, & Psionic sources had different labels, though at least some of each of them used implements and/or were subject to dispel magic.)
Now, it's true that Fighter (EKs) literally cast spells in 5e, and that the 5e stab at a non-spellcasting Ranger merely re-fluffed spells as not-magical. But that's 5e.
So, what, your actions revolve more about supporting others to be more effective in combat? Similar to a bard or something?
Mechanically, a bit, yes. "Leader" Role classes in 4e, which included the Cleric, Warlord, Bard, Shaman, Artificer, and Ardent, tended to have attacks that were just OK, as attacks (no one was really
weak in combat, unless very specifically designed to be), but also helped allies out in one way or another, typically restoring hps, granting temporary hps, granting tactical movement, or 'buffing' in a variety of forms. The Warlord and Shaman also had options that outright used their action to grant a (usually enhanced) action to an ally.
A 5e version would be stepping into the same shoes as the 5e Cleric, Bard, Druid, or, at the outside, Paladin. Which is saying something, as those are some of the most versatile/potent classes in the game. The support role in 5e - which puts a lot less healing resources in the individual character than 4e did, and which is tuned to a
long day requiring a lot of support resources to get the party through - requires a lot of versatility and a resource-heavy class chassis, as a lot of that 'power' is going to flow to the rest of the party, rather than glorify the support character, itself.
No sub-class hung on the sparse, tanky-DPR fighter chassis could pull it off (the Paladin gets there thanks to /both/ spells
and silo'd/dedicated supernatural healing and support special abilities), the BM, with a handful of minor support maneuvers, and the similarly minor dedicated support abilities of the PDK amply demonstrate that lack.