Probably a toss up as to which is cheaper, driving or transit. If you don't get slammed with traffic, something that probably would happen, then driving might be faster. However, many people find driving around NYC very trying. And where you're going isn't easy to get to from Newark Airport.
First the car issues: If you want to drive across Manhattan, then that is cheaper, however getting across lower Manhattan can be very interesting. Toll for Holland tunnel $6, Manhattan bridge free. Parking in Brooklyn no idea. My guess at least $15 depending on how long you stay.
If you choose not to drive across Manhattan, then add another $9 for either the Verazano Bridge going via Staten Island, or $9 going via the Brooklyn Battery tunnel (you'll drive downtown on West Street). You'll still be paying the 6 bucks to cross the Hudson.
Gas, figure at least half a tank, so that's maybe another 15 bucks. Allow at least 1 hour to 1-1/2 hours for travel time one way, maybe more depending on the time of day, route taken, and how good you guys are with reading a map and translating it into reality on the steets of NY.
My question, why are you renting a car? The hotel provides a shuttle to/m the airport. Are you going other places too? The trains won't be cheap, but if you are renting the car only for this, then the trains may be cheaper overall. If the car has other purposes, then driving may be better.
Now for transit: Hotel shuttle to airport airtrain, not really a train either, it's a monorail. Airtrain to Newark Airport train station. 15 - 20 minute ride, no idea on hotel shuttle.
NJT train to NYP, fare $11.55 one way each, round trip total $46.20. Trip time 1/2 hour. NYC subway #3 train to Utica Ave, 2 bucks per person one way, round trip $8. Best guess, 45 minutes to 1 hour. Then either walk .7 miles south (approx 10 blocks) or take a #46 bus on Utica Ave (free transfer with Metrocard).
You can also take the #2 train, however do not get off at the President street station. That's too far away, get off at the Winthorp St station, then walk .9 miles east. (approx 13 to 14 blocks). Nearest bus is the #12, 2 blocks south of the station, but even then the bus misses the hosptial by like 4 blocks or so. Again a free transfer to/from the bus.
I know nothing of these neighborhoods, so I can't speak to safety either walking or parking your rental car. I'm also not real sure just how close the Mariott is to the nearest airtrain station. But I do know that Mike H. has stayed there and he always took the shuttle. Seemed like he said it took 10 minutes, assuming that the bus was right there and not already on a run.
Ps. NJT train schedules can be found
here. The airtrain probably runs every 10 to 15 minutes, maybe less depending on the time of day. More info
here. NYC bus and subway info, maps, and schedules can be found
here.