As an aside, the work around showing P and NP being different or the same is a great example of the scientific method at it's best: all the experts generally agree that P isn't the same as NP, but that doesn't stop them from going through any proof which concurs with that belief with a fine-tooth comb. There is a such a clear separation of believe and the need for formal proof that there's a $1 million dollar prize available for a proof that shows P is the same as or different to NP.
Given that few of us are likely to win that prize, why should everyday people care about if P is the same as NP. The simple answer is that it is applicable to many everyday problems, such as packing a bag. NP stands for 'nondeterministic polynomial time' which is a very complex way of saying that, when given a solution to an NP problem, you can easily show it is a correct solution to the problem. In the case of packing a bag this is as simple as seeing that they've fitted all the items into the bag.
A problem which is 'P' is a 'polynomial time problem', which is to say that finding a solution from scratch is as easy to find as showing that a solution is correct to an NP problem. So if it was to be shown that P problems are not the same as NP problems then it would confirm that it really is sometimes easier to solve a problem when you've got the solution.
On the other hand, showing that P isn't the same as NP doesn't help practical everyday computing. The vast majority of polynomial time problems still take too long to solve with current computing technology, so all a proof would show is that there are very very hard problems as well as very hard problems. Most importantly, showing P isn't the same as NP certainly won't stop your word processor crashing when you've forgotten to save for 10 minutes.