Turing Completeness is based upon the ability to simulate a Turing Machine (if/else and random access to infinite memory). Obviously there is no such thing as infinite memory, but that is a nitpick that we don't pay attention to when talking about Turing completeness.
That we can create a physical system that is Turing Complete (a computer) in our physical universe strongly suggests that our universe is Turing Complete.
The initial question was about whether the universe was Turing Equivalent. A TE system must be TC, with the additional restriction that it has no more capability than a TM. According to Wikipedia, all known Turing Complete structures are Turing Equivalent.
However, as there is much about the universe we don't know, it is possible that there are some functions of it that are not simulate-able on a TM, and thus while the universe is TC, it is not TE.
So the universe is probably Turing Complete.
All currently known Turing Complete systems are Turing Equivalent.
This may suggest that the universe is Turing Equivalent, but answering that question is for smarter people than I.
The obvious consequence of the universe being Turing Equivalent is that as the universe can simulate/sustain a physical computer, so can a physical computer of whatever complexity simulate a universe. The simulated universe must be less complex than the one which is simulating it, as a universe has only a particular amount of information in it (so far as we can tell - size of the visible universe and all that), and only some subset of that state/information is available for simulating universe-b, each simulated universe is inherently limited by the amount of information in the universe 'higher' in the chain that is used to simulate it.
If all information in universe-a was being used to simulate universe-b either:
1) Overhead from the simulation would cause the limitation effect.
2) There is no overhead, universe-a perfectly simulates universe-b with no overhead. In this case, they are equivalent and the same.