It should be different if it is a dwarf hero that is solving all these quests, or that it's a blood elf hero and not a night elf hero that helped out the nightborne - and while the story can still go exactly the same way, because afterall, it is really the NPCs that affect, we can at least get something to show for being a race.
Take magic, Dalaran might seem to be the wonder and hold the secrets of magical knowledge to astound everyone - except if you're a blood elf, or you're a night elf highborne or nightborne (if they become playable), and it should be reflected.
Not everyone's human or orc and it should show. How many times have you read quest text and wondered - hang on, doesn't he know I'm a Blood elf, or I'm actually a ... ?
I like how you gather order resources for your class order, what if they did the same things for race as well? In addition to doing stuff for your Order Hall, you also have your Capital City campaign and do stuf for your race. And it can influence the story in interesting ways, for e.g. the horde NPC toons and warriors will be dominated by your race's troops.. so no more seeing orcs all the times or humans, if you're a blood elf, you'll see Blood elves, if you're a gnome the Brokenshore would have been full of GNome troopers.
Go to a place like Suramar - you will hear the Blood elf perspective..sure the nightborne and night elves would be the ones you're ddealing with and you'd hear their side, but then you will have blood elves coming in to help, and they'd be giving you quests concerning the nightborne - if the story has the nightborne going towards the night elves, you could be doing interesting things like sabotaging it even while trying to help them, if not you could be trying to persuade them to ally with you instead..
Long and shot, your race would matter. What if you were a troll? You would be hearing a lot more about Vol'jin and his replacement and how the Troll view all the night elf stuff you experience in Suramar, Azsuna and Val'sharah - would it be scorn to see what happened to the nightborne? would it be sympathy? You can't help but think a goblin would be angling for rewards and a gnome would want to retrofit gnomish engineering to help the elves... all sorts of crazy potential.
While it might be too late for Legion, it's certainly something they can do for 8.0 ... can you imagine? With sub-races in the picture, all the race insight you would get on everything for being a race.. and it doesn't have to detract from the main story at all, what can happen can still happen, but you can have your race play an integral role or at least see it's reaction to what you experience.