To what extent is an x86 machine equivalent to a Turing Machine?
The difference is: since x86 machines are finite, Turing machines can decide languages (decision problems) that cannot be decided by any x86 machine.



As I explained before, the idea of 'the set of decidable computations' is a category error.  Decidability is a property of formal languages (or equivalently, of decision problems), not of computations.  So, no, the statement in the second paragraph of your question is not correct.  It's not even wrong -- it is neither true nor false.

I suspect that perhaps you have found a statement of the Church-Turing thesis that is worded in a way that is confusing.  I suggest reading a standard reference on the Church-Turing thesis, and don't rely solely on the wording you have bolded.

It is true that every decision problem that can be solved by an x86 machine can be solved by a Turing machine (or, equivalently, every language that can be decided by an x86 machine can be decided by a Turing machine).  This is true because you can program a Turing machine to simulate the behavior of an x86 machine.  However, the converse is not the case: any real x86 machine has a fixed and finite amount of memory, while a Turing machine can use an unlimited amount of storage on its tape, so there are decision problems that can be solved by a Turing machine, but not by an x86 machine.  (Even if we take into account the amount of storage provided by disks, disks use fixed-length addresses, so there is a fixed and finite upper bound on the maximum amount of disk storage that can be addressed from a standard x86 machine.)

If you wanted an x86 machine to be equivalent in power to a Turing machine, you'd have to provide some way to extend its amount of storage without limits.  If you had a way to do that, then yes, any decision problem that can be decided by one could be decided by the other: each one could simulate the other.