The difference here is that with busy beaver numbers, e.g. BB(101) we can, presumably, write their values with our notation; it's just that we often cannot prove that any particular value written in our notation does indeed denote the value for that function. So if we write 100000...0000 with an unholy number of 0s there, it might be the value of BB(101), in particular we might not be able to prove that it isn't.
On the other hand, for a non-standard number, c, it is definitely the case that 100000...0000 is not c, because, whatever c is, it is strictly greater than 100000...0000, or any other number we can write down.
And thus when it comes to proofs about the termination of Turing machines that do not actually terminate, the unsound system is claiming that some machine terminates, but it doesn't terminate in 1 step, nor 2 steps, nor 3 steps, nor ... nor 100000...0000 steps, nor 100000...0001 steps, nor 100000...0002 steps, nor .... However, regarding the BB(101), the (presumably) sound systems we use such as PA, or ZFC, do not claim that BB(101) isn't 100000...0000. They just may not be able to prove anything one way or the other.
The difference here is that with busy beaver numbers, e.g. BB(101) we can, presumably, write their values with our notation; it's just that we often cannot prove that any particular value written in our notation does indeed denote the value for that function. So if we write 100000...0000 with an unholy number of 0s there, it might be the value of BB(101), in particular we might not be able to prove that it isn't.
On the other hand, for a non-standard number, c, it is definitely the case that 100000...0000 is not c, because, whatever c is, it is strictly greater than 100000...0000, or any other number we can write down.
And thus when it comes to proofs about the termination of Turing machines that do not actually terminate, the unsound system is claiming that some machine terminates, but it doesn't terminate in 1 step, nor 2 steps, nor 3 steps, nor ... nor 100000...0000 steps, nor 100000...0001 steps, nor 100000...0002 steps, nor .... However, regarding the BB(101), the (presumably) sound systems we use such as PA, or ZFC, do not claim that BB(101) isn't 100000...0000. They just may not be able to prove anything one way or the other.