And, perhaps more importantly, the user sees your app and branding in the end and they can choose to pick you again. Which is not the case for uber either.
Not sure about California, but here in France it's the kind of difference that would change the nature of the relationship. Although it would be evaluated as a whole, and would be under a lot of scrutiny to determine whether this change wasn't meaningless for practical purposes.
I suspect that in practice there seems to be a political consensus in Californian that Uber is evil. The rest is just trying to find ways to punish them.
If the existing laws can be interpreted to punish them, they will be used. If not, the coalition against Uber will look into getting new laws passed. (As they already have.)
Thus rendering any discussion about whether existing laws can justify punishment somewhat moot.
(Of course, even a moot discussion can be interesting. Just like people can have fun arguing whether the Stark Trek Enterprise would beat a Star Wars Destroyer.
The legal discussion is also very important to Uber lawyers. Even if they lose in the end, any delay is worthwhile.)
I think giving the ability for one driver to promote their services over other drivers is also a key feature. Each driver needs to be able to differentiate themselves and not be essentially interchangeable.
That's the a key difference for me to consider it a contractor/client relationship rather than a employer/employee relationship. I'm not a lawyer, so I don't know what the law says in each country Uber operates.