Incredibly close so far. They've changed a few rules of the fungus, in the game you can be infected by airborne spoors so they had masks sometimes, the proximity thing doesn't exist, and the fungus doesn't move around like that last scene where it comes out of his mouth.
The only big story thing I've noticed so far is how that last scene played out. Pretty sure in the game it's FEDRA that kills all the fireflies and Tess stays behind to stall them long enough for Joel and Ellie to escape.
And honestly I think both of these changes make some sense.
The other change is that in the original story, Joel and Tess weren't originally trying to get a battery, nor was Joel really looking for Tommy at that point. They were screwed by Robert on a deal for weapons so you go after him and find out he sold those weapons to Robert instead (Marlene shoots him in the face after he gives up the info). This leads you to Marlene and the fireflies, who promises you more weapons if you help smuggle Ellie out of the QZ to the fireflies at the State House, who were going to get her west. Just as in the show, you find them all killed. This is where Joel decides to find Tommy, because he was a firefly and may know how to locate the group.
In the game, the car\battery comes up later when you end up at Bill's, he helps you get a vehicle and you end up needing to acquire a battery for it.
So they kind of condensed those two stories into one, and have changed things a bit to make Joel\Tess a bit more sympathetic.