A proof of P = NP would have huge practical consequences. A proof that P != NP "only" represents a huge advance in the theory of computation.
Could you explain what some of those consequences be? Does P = NP really need to be proved to achieve those practical consequences? Can't it just be assumed to be true and then see what the result is?
Wikipedia should answer most of your questions regarding the consequences:
NP problems can be solved, it's just slow (think brute force method, running through all the possibilities).
http://en.wikipedia.org/wiki/P_versus_NP_problem#Polynomial-...
The first historical example of an important proof which was non-constructive was Hilbert's Basis Theorem in 1888, which solved a famous problem posed by Paul Gordon. It is now viewed as in some sense being the first salvo in the debate over constructivism in the 20th century. See http://en.wikipedia.org/wiki/David_Hilbert#The_finiteness_th... for more.
NP questions take a long time to answer.
P questions take much less time.
The statement "P = NP" means that any NP question can be transformed into a P question. Answering the P question gives you an answer to the NP question.
So, if P = NP, then we can answer questions that we thought were slow, much more quickly than we would have thought possible.
Proving P = NP would (presumably) give you that method for transforming NP problems into P problems. So you would suddenly be able to figure things out easily that currently are hard and take a lot of computing power.
It's often assumed that P = NP would make cryptographic problems much easier to break, but apparently that's not really the case: http://world.std.com/~reinhold/p=np.txt
(Though, as you say, there could still be an O(n) algorithm - this is not known).
What I said is true even for linear programming (a P-complete problem) - it has polynomial and efficient algorithms - though the latter are actually not polynomial in the worst case. :)