What if WoW post vanilla never got new races to play. No blood elves, no Worgen, no allied races. Just the 8 core races we were given at the start and instead they added more and more customization options for these 8 every new expansion?
How different would the game be? Story differences?
For me, the game would be a little different because I have friends who play those races. But I would keep playing these cute little gnome guys! 
I dunno if I could play without Belfs.
I started as a Night Elf, I assume we might still be here anyway.
Who is Kaznarok?
if by extra customization options you mean i can choose to keep my old UD casting animations then i’d be on my knees in a second
The OP. I guess this is a thing he does maybe?
On topic: I guess it would depend on what you mean by “customization options”. Would we get sub-races and so forth, like breaking out the Wildhammers and Dark Irons? Or Forsaken getting to use other undead models.
I’m sure the story differences would be pretty extreme. If they hadn’t had to make Draenei playable they probably never would have bothered combining them with Eredar and they’d still look like weird swamp gremlins. They probably wouldn’t have killed off Kael’thas immediately so that the BE could keep being antagonists forever.
Worgen would still just be monsters for us to kill, goblins would still be neutral NPCs that ruin the environment.
No playable Pandaren would have taken a lot of the magic out of Mists, and life in general.
Void elves wouldn’t exist at all. Maybe this is starting to sound better…
This 
Without going into too much of it because Briselody touched upon some key parts. There were already a few major retcons from WCIII to where we are currently in WoW. The retcons weren’t necessarily bad and just for example, Suramar, the Broken Isles and the entire Legion expansion wouldn’t exist.
Also, this, the elegant Draenei Females you see before you, Kaznarok would look similar to this
Yeap, a face for radio.
To answer your question, I believe the story would stay focused more on Horde vs Alliance and if we didn’t get additional races then I could see a point in time where we get the original request of sub-races as customizations, like the Revantusk (Forest Trolls) joining the Horde.
Horde would look pretty empty without blood elves, just sayin’.
I seem to have acquired an infamy for making incredibly stupid topics.