Isn't that the wire shown at the bottom right corner of the first diagram ?
I think the two power supplies have to have a common ground.
The first diagram wasn't there before: it's a later edit.
The 298 datasheet doesn't use the term Vcc, so that Vcc might be either Vs the motor supply, or Vss the logic supply, who knows?. According to the L298 datasheet Vss is actually supposed to be between 4.5 and 7, with 5 typical. That 3.7 to 7.4 is a little outside that, but I see they're using Lipos so that's where that comes from: either 1x for 3.7 or 2x in series.
So back to the original question: You can drive motors with up to almost 50V (Vs max = 46). Put about 5V into Vss to drive the chip (assuming their experience shows 7.4 is safe, ymmv) and put enough voltage into Vs to drive the motors. The whole idea of the 298 and similar chips is to let you control that hefty motor power by "tickling" the chip with low power controls from somewhere, Arduino in this case.
A word on the efficiency of the 298. Look at the datasheet, find the line VCE Sat at bottom of page 3/13. It shows a minimum voltage drop of 1.8, and this becomes 3.2 or 4.9V at 1 or 2A. You will therefore always need to make Vs at least 2V higher than the voltage required by your motor; you'll need it to be 5V more at high current. That's why the board has the chip mounted on a heat sink as big as a truck: lots of energy getting wasted into the air as heat.
You might like to consider a more modern approach like this, tiny and no heat sink, although that particular model doesn't go quite to 12v.... Point is the 298 is "old school" and very inefficient: check the supplier sites and you'll find a less wasteful chip with the right voltage. (Edit: This one handles 15v, and is 0.8" (20mm) square.)
EDIT: You might like to give some thought to that 3.7 to 5V that they're putting into the Arduino. Have a look at the Arduino UNO specs or Leonardo specs; nowhere does it say that they can run under 5V. In fact, 5V will only work if you put it into the USB socket; other inputs like the barrel jack or Vin require 6V min, 7V and up recommended. So what you could do here, assuming as I said that 7.4 is ok for the 298's Vss, is use 7.4V to drive the 298 logic on Vss, the motors via the 298 Vs, and the Arduino via Vin.
Then if you need higher voltage for the motors, leave the 7.4 for Vss and the Arduino, put the 12V into Vs for the motors. Link the grounds or else you won't have a circuit...
More edit: I'm leaving for work soon, it's 615 am here.... my office lan is intranet only, and I get almost non-existent mobile modem coverage, so I'm off the air for the day, most likely.