It’s probably a toss-up. Here are some factors to consider.

1. Most anything would be cheaper than an Eastern European railpass. I’d buy point to point tickets for these trips.
2. Petrol is 3 to 4 times what it costs in central US. Gas is $3.27 a gallon today at my home, so I’d figure $3.27 per liter (3.7 liters = 1 gallon) in Europe. Parking is expensive as are toll roads although you may not have many toll roads to deal with. Having a car would be a nightmare in any of the cities you mention. Also make sure you figure drop off fees (sometimes renting in one country and dropping off in another doubles the cost), extra charges for each country you drive in, insurance, extra charges for more than one driver, extra charges if you’re under age (21 in some cases, 25 in others) and VAT in your calculation. These may be included in the quote you’ve got, but may not be.
3. IF you’re small enough and pack light enough to cram 3 of you into a sub-compact, standard transmission car, you will about break even with taking the train. Rick Steves rule of thumb is $150/per person per week for train, $500 per week total cost for a car. With 4, you will probably save money renting a car – assuming you don’t need too big and luxurious a car to comfortably accommodate your party.
4. Since your overly rushed itinerary only includes big cities, IMO you won’t save enough to give up the comfort, convenience and general lack of hassle of the train. Train is generally faster, (city center to city center) and more comfortable. With dozens of trains each day to choose from between these cities, you give up very little in terms of flexibility by taking the train. If your timeline is more reasonable than what’s in the trip planner and you intend to visit smaller villages, explore rural areas and stay in suburbs as opposed to central cities (adding some time and expense for local transportation to get to what you want to see/do in the cities), then driving would be better.
5. You won’t have any trouble driving and it’s relatively safe. Consider driving a full time job for 1 person and navigating/spotting (looking for road signs, parking spaces, where you need to turn, destination signs) to be 1 and 1/2 full time jobs to be shared by the rest of you. Highway fatality rates in this part of Europe are significantly higher than in the US, but you won’t find it scary to drive in this area — other than the nightmare of driving in the central part of the cities.

Would it also be cheaper than point-to-point tickets? Maybe yes, who knows.

More lively than in the U.S., you have to pay much more attention. Especially when driving in Poland, that’s the region’s nightmare. But other than that, EU roads are about as safe as US roads.
BUT: Are you familiar with driving with a manual gear shift? Are you familiar with European roadsigns? Do you own an international driving license? Do you count on paying the highway toll – CZ, PL, SK, HU and AT all have it in a form of a window sticker, valid for AT LEAST two weeks or a month, no single-use payments in booths. Or do you resign from using the highways and thus crank up your consumption? Take all this into consideration.

Definitely won’t save time Prague to Krakow (bad roads, good rails) or Vienna to Munich (good roads, but excellent rails). It might speed up Krakow to Vienna, but there’s a direct night train on that route anyway where you can sleep over the whole trip. There’s a night train from Prague to Krakow, too. And there’s a high speed service from Vienna to Munich.

European cars’ consumption is stated in litres per 100 km, unlike the USA, where you state the distance driven on one volume unit.
The modern cars with good motors offer around 6-7 litres of (ideal) consumption per 100 km at 90 km/h. The consumption in the city ranges at around 8-10 litres per 100 km.
One litre of fuel costs about €1.40, which is about US$ 2.05. (By the way, that makes US$ 7.80 per gallon).
Prague to Krakow is approx. 700 km. You will burn at least 55 litres if driving modestly, that means at least €77 or $112 on fuel alone, without tolls and without the borrowing price itself. Train seat can be bought for €19 and a sleeper for €29 in advance.
Anyway, look at the consumption of a car you’re choosing. 6 liters not bad, 5 liters great, 4 liters unbelievable.