How come x^2 = 25 has 2 solutions but x=root(25) only has one? Aren't they the same thing?

(This is something that I didnt fully understand for quite a while at school.) So when we are solving x²=25, in order to get x "on it's own" we square root both sides. However the definition of root(x) is root(x) = |x| so once we square root our equation we get |x| = |5|, since 5>0 we see |5| = 5 so our equation becomes |x| = 5. From solving modulus equations we know the easiest way to do this is to consider two seperate cases, one case when x >=0 and a second when x<0. This leads to us getting 2 solutions, which are x = -5 or 5. For x=root(25) we dont have to square root both sides so we just end up with x = |5|, again since 5>0, |5| = 5 so x=5. So the second equation (in the queston) has one solution but the first equation has two.