Others have covered the downside of registering with the state. The future will tell us how bad that will be.
You can take CCW training and never apply. I've recommended that to a few people, because the cost is reasonable, and IMHO, the training is very helpful.
If you do get the permit, as noted above, it is easier and quicker to buy a gun. That is not a small consideration, especially with the BS being pulled on us lately by the Fed's fooling around with the NICS system. This is very, very helpful, .... assuming, again, that you are willing to register that purchase with the state (via the 4473).
Another advantage is if you get some LEO who has a hair up his ... pants. Perhaps your gun is in the wrong place in your vehicle, your shirt falls over the gun or there is some other cute way he / she can harass you (most don't, but if you get a rookie, or you're young, then you might get the wrong sort). If you've got the permit, then it makes it tougher for them to charge you with one of these BS victimless 'offenses'.
I decided to get a permit. There are two ways to go generally, in life ... keep your head down and your mouth shut, or expose your beliefs and argue your convictions. I chose the latter, mainly to try and persuade others of the logic and truth of the RKBA. Time will tell if that was a wise choice.
Good luck with your decision.
Regards from AZ