How is the Time Complexity of a Function Problem different than its corresponding Decision Problem?
The Wikipedia article is confusing.

There are two things going on here.

Here's first thing: the complexity class P is defined to be a class of decision problems.  Thus, by definition, no function problem is in P.  So it's correct that the decision problem for gcd is in P, but the function problem for gcd is not.  Of course, this is pretty trivial.  So the first thing is not very interesting -- but it does explain why the Wikipedia article makes that odd comment about the decision problem for gcd being in P.

The second thing is more interesting: is there a difference in the running time between the decision problem and function problem?  Sometimes there is.  For instance, consider factoring.  A function problem is: given a number $t$, factor $t$ into a product of primes.  A decision problem is: given a number $t$ and primes $p_1,\dots,p_k$, check whether $t = p_1 \times \dots \times p_k$.  There is no known polynomial-time algorithm for the function problem, but there is an easy polynomial-time algorithm for the decision problem.  So, it's apparently easier to check whether the answer is correct than to find the answer.  Indeed, for any NP-complete problem, we believe it is easier to check the answer than to compute it, so the function problem appears to be harder than one plausible formulation of the decision problem.  (For that reason, when we turn function problems into decision problems, we usually do it in a different way than the one you mention.)

What about for gcd?  The function problem is: given $x,y$, compute $\gcd(x,y)$.  The fastest algorithm known for this takes $O(T(n) \log n)$ time, where $n$ is the number of bits of $x,y$ and $T(n)$ is the time to multiply two $n$-bit numbers.  It is an open question whether this can be done faster than that, or exactly how fast this is.  A plausible decision problem is: given $x,y,d$, check whether $\gcd(x,y)=d$.  Here it is not known what the fastest possible running time is.  You can check whether $d$ divides $x$ and $y$ in $O(T(n))$ time, so you can check whether $d | \gcd(x,y)$ in $O(T(n))$ time -- faster than we know how to compute the gcd in the first place.  But I don't know of any algorithm to turn this into a way to check whether $d = \gcd(x,y)$ asymptotically faster than computing the gcd from scratch in the first place.  So we just don't know in this case whether the decision problem can be solved faster than the function problem.