Matt,
You've posed a good question. Most people would've asked the boring same-old, like "which kit is best" and "what do you like about yours"... instead your question makes us think and forces us to be objective. Smart man...
Let's see... some cons...
1. Gas Mileage
Now, this might seem like a "DUH-HUH" answer, but you'd better be ready for it. If you choose a killer motor like I did, you WILL be stopping at the fuel pumps a LOT. I'm not whining about it (small price to pay for the fun), I'm just warning you. I get about 6 MPG on average... and that's not the
cheap gas, either.
2. It Ain't Your Honda
Remember: This is a 1965-technology car. Yes, my replica rides just fine, but if you're expecting a soft, cushy, comfortable ride on a regular basis you'll be in for a very sharp learning curve. As mentioned before, this car can be a brute... and, as also mentioned before, this is both a pro and a con, depending on your mood.
3. Attention Getter
If you'e not in the mood to spend 15 minutes at the gas station talking to Mr. Billy B. Local about the Falcon he had when he was in high school, you better not pull the Cobra out of the garage. The car does get a
tremendous amount of attention. If you're in the mood, it's wonderful. If not, well...
4. Noise
I absolutely love my pipes. For all intents and purposes, they're wide open. But they
can become tiresome to listen to after a while (again, your mood will be the judge). Also, they
can cause tension between you and your neighbors. This hasn't happened to me personally, but others on this site can tell some stories. The car has a VERY high likelihood of grabbing the attention of Capt. Paul O. Ficer who's been able to hear your car coming for the last two miles...
5. Not Lockable
This is simple: Don't leave anything in the car that you're not willing to have stolen. This hasn't happened to me yet, but it's just a matter of time, I'm sure.
6. Did I mention gas mileage?... I did?... Good.
7. Upkeep
Again, we're talking 60's technology here. You should expect to spend more time than usual re-adjusting your valve lash or tweaking your timing or replacing your fouled plugs. Almost none of which you would EVER have to do for a daily driver. I enjoy it, myself, so it's not so much a con for me, but people are different.
8. Sunny Days Only
Now, I
KNOW some of the CC guys are gonna give me some flack for this one...

it's a long story... anyways...
Be prepared to have the car sit alone in the garage during those weeks when you REALLY need to take it out just to unwind and relax, but instead it's raining, so you have to wait. Or worse yet, you get poured on while you're out somewhere.
Now... with all these cons, would I prefer to have something different?
HECK NO! I prefer what Ferris Bueller said...
"Oh, and about the car... it is SO choice. If you have the means, I highly recommend picking one up."
======EDIT======
9. How much did that cost ya?
I can't begin to tell you how many times I have been hit with this question. Surely there is no other car that garners this question more than a Cobra. It can be VERY annoying to have someone ask this, especially right in front of a co-worker or friend. I handle it differently every time, but I always try to be nice and cordial, even if the question is not. Sometimes, the question doesn't bother me if I
know that the individual is seriously interested in acquiring one himself or herself, but that is rare.
10. Big Brother (and everyone else) is watching you.
You need to be "on" at all times. It's almost like you're a celebrity or something. Don't snap at anyone. Don't give someone the finger in traffic. Don't cut someone off. Don't cuss. Don't adjust your crotch. Don't pick your nose. Don't... oh you get the idea.