Equivalence of Turing Machines
These arguments are not really related, on several levels.

The first is that existence and computability are not the same thing. That is, even if there exists some object (e.g. a TM), it does not mean that finding (or computing) such an object can be done using a TM. 

However, there is a more basic difference between the two statements. The first says that for every TM $M_1$, there exists some equivalent machine $M_2$. The second statement is that deciding, given $M_1$ and $M_2$, whether these two specific machines are equivalent, is undecidable.

Just to emphasize the difference, consider the following problem: given a TM $M_1$, is there some TM $M_2$, that is different from $M_1$, which is equivalent to $M_1$? This problem is trivial - the answer is always "yes" (e.g. just add a redundant state to $M_1$).