Could the current solution to Fermat's Last theorem be the solution that Fermat 
derived?  If not, why?

The Comments posted below summarize much of the prevailing view on your 
question. A good detailed history (for the mathematically literate!) of 
Fermat's Last Theorem can be found in the article "Fermat's Last Theorem," by J 
J O'Connor and E F Robertson, February 1996, which can be found on-line at

http://www-groups.dcs.st-
and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html

They note that "it may well be that Fermat realised that his remarkable proof 
was wrong, however, since all his other theorems were stated and restated in 
challenge problems that Fermat sent to other mathematicians. Although the 
special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know 
how to prove these) the general theorem was never mentioned again by Fermat."

The article goes on to describe the subsequent history (Fermat's famous 
marginal note is thought to have been written around 1630) through the solution 
of Wiles which he worked on intermittently from when he first became fascinated 
with the theorem at the age of 10 in 1963. He was finally awarded the Wolfskehl 
Prize (unclaimed for almost 90 years) in 1997 for his solution. See the 
European Digest, Vol. 1, February-March 1998 for a discussion of the prize:

http://www.european-digest.com/ecd01/docs/digest18.htm

While it is difficult to be definitive--no one knows for sure what Fermat did 
or didn't know, there seems to be near-universal agreement that

1) Fermat probably did not really have a complete solution to his theorem,
2) The solution of Wiles is very unlikely to have been very close to whatever 
ideas Fermat did have.

Many additional sites discuss Fermat's Last Theorem. These can be found simply 
by doing a Google search on:

"Fermat's Last Theorem"

No. The current solution runs to hundreds of pages, whereas Fermat referred to 
a simple and elegant solution. For example, see http://www.european-
digest.com/ecd01/docs/digest18.htm

"Wiles' solution, though, is not Fermat's solution. Though Fermat had said the 
margin of the book was too small to contain his remarkable proof, he did write 
to a colleague that he had devised a relatively simple way to solve it. Wiles' 
solution, on the other hand, runs to over 100 pages, is complicated and uses 
techniques not known in the 17th century."

It is also possible that the solution Fermat had in mind might have seemed 
correct to him at the time, but it may have contained errors that would have 
been exposed if he had released it to the public.

And, given the copious research done on this problem, the latter seems much 
more likely.

Remember that we have only Fermat's comments to support the claim that he did 
indeed solve the problem. Although it is extremely unlikely that the current 
solution is the same one that Fermat conceived, it is impossible to prove that 
it is not. Remember, also, that we have no knowledge of his ever actually 
deriving it (putting all the steps to paper), let alone having conceived it.

It's impossible for the current solution to have been Fermat's
solution. Wiles uses a lot of "Galois Representation Theory" and ties
together two areas of matheamtics that were only conjectured to be
similar. All stuff that has been developed in the last 20 years or so.
Fermat could not have done that because he was missing about 300 years
of mathematics development.

While the case n=4 was finally proved by Fermat himself, he was not
able to solve the problem for other exponents. The case n=3 was solved
by Euler.

It was not uncommon for Fermat to write conjectures as if they were
theorems. For instance, he stated in 1640 that the expression
2^(2^n)+1 is prime for every n>0. But this is obviously incorrect,
because 2^(2^5)+1 = 4294967297
 = 641 x 6700417. These numbers are now known as Fermat numbers and
you can see their known factors at:

http://www.prothsearch.net/fermat.html

Can't someone just feed the problem in to a computer, and let it do the working out?

>Can't someone just feed the problem in to a computer, and 
>let it do the working out?

Nope. Computers can't find proofs.
If you mean checking the FLT for a single case where it breaks down,
well, people have tested the equation it up to large large numbers
without success... Knowing full well that the upper bound is infinity,
it was an exercise in futility.