I'll put in my $0.02 with the disclaimer that I know next to nothing about drag racing - my thoughts come 100% from the perspective of road racing. In the GRAND-AM Continental Tire Challenge Series, the Challenger 392 will indeed compete directly against the Boss 302R Mustang, so I'm pretty familiar with some of these issues though I'm not an expert by any means on either car.
Engine - Advantage Challenger:
This one isn't as simple as it the numbers might suggest. 6.4L HEMI = 470hp/470lb-ft, 5.0L Coyote = 444hp/380lb-ft. Obviously the 6.4 has a huge advantage in torque, and it is indeed torque that wins races (horsepower sells cars). But, the Mustang has a 7500rpm redline versus the 6400rpm limit for the 6.4. In road racing, it's really nice to have a wider RPM range to work with, so the higher redline is nice. But, at the end of the day, GRAND-AM is making us run a restrictor to keep our power in check, and they're not doing that for the Mustang. So, there you go.
Weight - Advantage Mustang:
The Challenger is nearly 4100 pounds as delivered, the Mustang is about 3650 (apparently - I have not weighed one). That weight differential is huge for the as-delivered cars and, honestly, is the only reason the two cars are comparable. For our racing program, we are reducing the weight of the Challenger to about 3300 pounds (no small task, mind you) to basically eliminate this variable. The Boss 302R has to run at 3300 pounds as well, difference being they will be running lead ballast to make weight, we are spending gobs of money to run lightweight this and carbon fiber that to have a chance to get to 3300.
Transmission - Advantage Mustang:
Both cars run Tremec TR6060 based transmissions. The Mustang unit, however, has some advantages - at least for road racing. First is the 2.29 gearset they run - very close ratio gears that are nice and evenly spaced all the way up. The Challenger gears are spaced much wider and there is a big jump from 4th to 5th and again from 5th to 6th making those two gears almost useless for road racing. So, with everything else (tires, rear gears, etc.) set per the rules, the Mustang is running 2nd through 6th gears and we can only use 3rd through 5th - effectively a 3-speed fighting a 5-speed. The other advantage the Mustang has is that the shifter mechanism bolts directly to the top of the tailshaft. The Challenger is a "semi-remote" setup with part of the shifter attached to the unibody and the other part attached to the transmission. As the drivetrain moves relative to the body, the gates move around and can be harder to find. This can be solved by stiffer (or solid) drivetrain mounts, but that has other consequences (wheel hop, harshness, etc.).
Suspension - Advantage...?:
This is a tough one because I really believe both cars have excellent suspension packages. The geometry on the Challenger is really good and the car handles better than it's size and weight should allow. The Mustang also carves corners really well, so in the end the difference (again, for what we're doing) will come down to who tunes the suspension better. There are a lot of variables and finding the sweet spot is a real challenge.
Exclusivity - Advantage Challenger:
Pretty simple here - I see about 1 Challenger for every 10 Mustangs that I see, maybe even less than that. The popularity of the Mustang is also its downfall in this respect - there are just a ton of them out there. Every Challenger I see turns my head, it takes a special Mustang for me to even notice.
So, there are some thoughts from somebody trying to really hard to make a Challenger (a little more than) equal to a Boss 302R in a venue where the Mustang holds many of the cards.