A few years ago, when I was teaching math at a big state university, a colleague told me the following.
She was comparing notes with a professor at a nearby school on how their respective real analysis courses were going. She told him that they had just proved that the square root of two was an irrational number. He laughed and said she was way ahead of him; his class had just proved that the square root of two is a number.
[Don't feel bad if you don't get it. This is not the sort of thing that normal people talk about.]
The joke was that, while it may sound impressive, showing that the square root of two is irrational is fairly easy. There's a nice, elegant little proof that is easy to explain and is suitable for anyone who has completed the first few sections of high school algebra. On the other hand, showing that a real number X exist such that X squared equals two is actually a bit of a challenge.
There is one other point here which ties in to our ongoing math curriculum thread. Namely that, with the borderline exception of high school geometry (and even there we cheat a little), a truly rigorous approach to lower level mathematics is wildly impractical. The order in which concepts are needed does not match up at all with the order of difficulty of proof (the fundamental theorem of calculus comes to mind). Therefore much (probably most) of what we tell students is back up with no more than a "trust us." We don't have to like this but we do have to acknowledge it.
This doesn't mean that proofs aren't important, but that the importance lies in the process and not in the result.