Relatively new to D&D, so sorry for not knowing much.
OK, quick version. D&D was the first RPG, and it was quite primitive, as one might expect. For 25 years it barely changed while many more-evolved new RPGs proliferated around it without any one of them ever gaining much popularity. D&D's publisher went belly-up and it was acquired by WotC, which made it's fortune on CCGs. D&D finally underwent some changes, on balance(npi), maybe for the better, WotC was bought out by Hasbro and turned even more corporate, there were some controversies, some miscommunications, some jerk moves on their part, and fans got uppity. In 2008, they rolled rev on D&D again and that new ed, among lots of other things that (arguably, in a technical sense) were a vast improvement over the old game, introduced one new class in the PH1, the Warlord. The fanbase fulminated with rage at being jerked around by a new edition again so soon, and 'the edition war' broke out between naive/new fans who were taken in by the shiny, new, much-improved system, and True D&D Loyalists who didn't want to see their beloved game desecrated by alien concepts like clarity, playability, balance or accessibility to new players. The latter won, prettymuch hands down, and 5e is the resurrection of True D&D. Minus, of course, the Warlord class, which had the misfortune of being introduced in 4e.
Honest, that's the short version.
Why is "Non-magic" healing important?
Because, during said edition war, huge battles were raged and grim atrocities committed to purge from D&D forever the unspeakable horror that was restoring hit points without magic (or weeks of rest, but mostly magic, because who waits weeks when you have magic that heals instantly and re-charges every day). Not to mention the arguably equally unspeackable horror of FIGHTERS CASTING SPELLS! XOMG. OK, really, the issue wasn't either of those things, it's that martial classes like the fighter and warlord were actually pretty nearly balanced with caster classes like clerics and wizards (I know that may sound weird, but don't worry, those days were over almost before they began).
So, 5e comes along and casters have far more than three spells again, and fighters and the like have only a couple of short-rest recharge powers that in no way compare. Everything's fine. But, there's non-magical healing in the game in the form of HD and Second Wind and a few other things here and there. Oh, and the Fighter does, indeed, cast spells. ;P So that's really kinda a non-issue, now, so long as any class that has magical healing is suitably and strictly inferior to casters in terms of daily resources (there's a loophole for short-rest-recharge resources) and general versatility and flexibility.
The Warlord, as a 4e martial class had a lot more versatility and flexibility and resources than 5e fighter, but less than a 5e caster, so it might seem like it could challenge the natural order. For some, 'non-magical healing' (or inspiration as hp recovery) is still symbolic of that issue. Given that there's already several types of non-magical healing in the PH (and SCAG), though, it's only meaningful in that symbolic sense.
What is the relevant difference between it and magical healing?
It's not magical. It could, in theory allow a party to operate effectively without a caster in the party, at all.
2) Remote-controlling allies sounds like
Yeah it does. What it doesn't sound like is what the Warlord actually did: granted allies actions. So the Warlord would use his action on an 'exploit' (a martial resource, like a BM's CS dice/maneuver - and maneuver is a much nicer name, isn't it, exploit just sound shifty) giving an ally a corresponding action, that it could use in a limited set of ways. The idea is that the warlord embodies the kind of leadership skills you see in a coach or really good boss (if you've ever been lucky enough to have one) or, ironically, even a plucky sidekick. So the Warlord shouts a warning or order or word of encouragement and the ally, warned/heartend/spurred-to-action gets to do something he wouldn't have otherwise, out of turn (on the Warlord's turn).
If you'd been playing D&D for a long time, and seeing how the game would talk about fighters being 'leaders' in some sense (in the classic game, a fighter who hit 9th level was called a 'Lord' (and a bit below that, a Warlord, coincidentally) and could claim a medieval feif, just like that - but he got zip in terms of skills that might help you actually run a fiefdom or organize/inspire soldiers to defend it. That trend continued all the way through until the Warlord was finally build around those skills (well, not so much the castelan skills, it's not like PCs become fuedal lords much anymore - in 1e, it was just expected).
So, yeah, no remote-controlling, that's left to Wizards who can cast Dominate Monster (or Magic Jar, is Magic Jar still around? I forget, there's been so many spells over the years.)
I really do want to understand the hype of Warlord
There's really no hype. There was the edition war, and a lot of vitriol and misinformation that still gets repeated.
(I wouldn't really want my character to be "inspired" by another character all the time, unless my character actually felt that way, but a couple times every now and then would be fine.),
The intraparty dynamics and RP are something the players involved can work out, if they feel the need.
but I usually see it described as more like the player of a tactician style game, where the other players are your units, and completely under your control.
Not how the mechanics ever worked...
...OK, except for one power in the PH1 that was errata'd as a result.
The whole magic thing escapes me still though. I do not see any true difference between healing and non-magical healing.
One's magic, one's not. D&D, for the first 34 years, had a slowly growing set of classes (it started with Cleric, and is now Cleric, Druid, Bard, Paladin, & Ranger) able to heal their allies with magic. Magic-use like that is actually pretty rare in genre, and, particularly when it was mainly the cleric that could adequately fill the role of party band-aid, a lot of folks didn't care for the idea of having a devout character be mandatory (in the 80s, and you may have even noticed, today, religion can be a fraught issue). So, after over 3 decades of being unable to have a party without someone mumbling and laying on hands, we finally got a class that enabled a party without a magic-using healer. That's great if no one (willing to take up that role, that is) at the table wants to play a religious type or an occultist entertainer, or if the campaign is trying to go for a low-/no- magic S&S vibe.
If I were to look at non-magic healing, I would want it to be more of a healer, with bonuses to healing over time, rather than instantaneous healing. Maybe a mechanic similar to the Song of Rest, or a regen capability like the Champion ability.
Part of the problem is even calling it healing. It gives you this idea that hps of damage are deadly bleeding wounds and getting back a few hps is making wounds magically disappear. Actually, in 5e, you can get all your hps back in an hour of rest by spending HD. The idea of Warlord 'healing' - Inspiring Word - was that it did what was no the tin - inspired an ally so that he could rally and keep fighting (characters could also do that on their own, with an action called, not coincidentally 'Second Wind' 1/encounter). So, not weird or crazy - or taking a long time like non-magical medieval medicine treating a severe wound might.