I played very little 4th edition, but I was there for the "great warlord debate" soon after the release of 5e. My understanding of it is this...
When making 4th edition they assigned power sources and roles to all of the character classes. Power sources were Arcane, Divine, Martial and Primal. Then the roles where Controller, Defender, Leader and Striker. They were able to fill most boxes with an existing class, but if one was empty they felt compelled to fill it. Thus the Warlord was born! (and a few others, like Swordmage and Warden).
The Warlord filled the Martial Leader role. The leader role (which I think was poorly named) is basically a support class that buffs or enhances the other characters in the party and makes them more effective. So the Bard was the Arcane Leader and the Cleric was the Divine Leader, for example. The Warlord could shout out commands and, as long as his fellow party members could hear him, were able to do things like take extra attacks or extra movement or recover hit points and things like that.
The problem with the Battle Master is that he is a Fighter with a few Warlord like abilities tacked on that can only be used occasionally. A Warlord in 4th edition could also be designed that way, but it could also be designed so that it was 100% support all the time and did next to nothing on its own. This came to be know as the "Lazylord" or "Princess" build and there are people out there that absolutely love it.
So hopefully that is an unbiased and not too controversial description of the Warlord and the difference from the Battle Master. I'm sure others can point out any omissions or inaccuracies.