should we drive or ride a bus to New York City?

Driving here, while difficult, isn't the real problem. PARKING is the big problem.

There is almost nowhere to park. And the few places that are available are prohibitably expensive.

That $200 you'd spend on bus fare doesn't seem so bad in comparison.

If you don't need the car when you get there, I would take the bus. It can be very difficult to find parking and when you do, it can be expensive. Most places in New York City have good public transportation.

There are hotels in Manhattan with free parking for guests. There's a reasonably priced one on West 42 Street. (About $250/night). If you can make reservations in one of those hotels, why not drive?

Heads up: do not even attempt actually to drive in Manhattan except to go to and leave your hotel.

Take Public Transportation.
Traffic is a real problem and Parking is a BIG problem
Very Expensive Parking.

The only thing stopping me from saying "ride a bus" is the fact that your journey will take 24 hours. Maybe for others it's nothing, but for me, that's a hell of a long time to be on the road. (You can read books or something to keep you distracted, but believe me, waiting that long would be agonizing.)

I guess ask yourself: With all other things being the same, what's more important for you, saving money or convenience? If spending any more than the $200 bus ticket is too much for you, then take the bus. On the other hand, if you don't mind paying upwards of $150 for three days of parking (not to mention gas and tolls), then bring your car. You can pace your trip a little slower knowing that you are not at the mercy of the bus schedule. Plus, you can make frequent stops if either your or your girlfriend needs to stretch your legs.

(I didn't count traffic here because [a] you'll hit traffic going to NYC whether you take a bus or bring your car, and [b] NYC is a city best explored on foot...once you get to your hotel, leave your vehicle at a parking garage for the entirety of your stay.)

You should definitely take a bus! I would suggest Peter Pan for the bus company. It might be a few more dollars but it's safe and worth it. If you wanna save a few more bucks you guys should take a bus/train around instead of a taxi or cab. But then again Uber does exist!
Have a fun trip!