Just buy one of each!
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Well, if you can only buy one...the choice of which car to buy comes down to personal factors and what you are looking for. I would rather drive the Z06 as a daily driver or in inclement weather. But those factors may not matter to you. If you are a big guy, you will find the Vette more comfortable. If you want kick-a$$ straight-line acceleration, the Viper is your car, though SoCal Rebell exaggerates the difference in 1/4 mile times between the two...Stock vs. stock there's about 1/2 sec difference. My GTS ran 12.3 stock vs. 12.8 for my Z06. I know there's more in both cars that I didn't tap into because I am not a seasoned drag racer. Great Viper drag racers can get 11.9-12.0 out of a Gen II Viper, and great Z06 drivers can get 12.4-12.5 (stock vs. stock). One fellow on the Corvette Forum even ran a 12.007, totally stock, including tires, but that sounds like an anomaly.
Also, for short bursts (e.g., stoplight-to-stoplight or 0-60), the Z06 matches the Viper. It is easier to launch the Z06 for sure, probably due to lower gearing. (Guys get 1.79-1.85 sec 60' times in Z06's regularly -- on the stock street tires.) My best 0-60 time (measured on G-Tech) is 4.20 sec for the Viper and 4.15 sec for the Z06.
On a road-racing track, I agree the Z06 seems a little more intuitive to handle. But there is also no question the Viper has similar handling limits, so in the end it's a matter of how well you learn the cars. Also, I'm a novice for road-racing, so perhaps my experiences should be taken with a grain of salt. For sure the Z06 has much better braking (at least compared with my '97 GTS -- perhaps this is different on the new ABS cars).
Perhaps the biggest difference between the two is gawk factor. The Z06 is just another C5, of which there are a lot on the road. The Viper is rare and defintely more breath-taking in styling. Whether or not you personally like the looks of the Viper better than the C5, the Viper will always be the one attracting crowds wherever you take it.