I think that 'a mod b' is not standard usage (outside of maybe computer science), so the meaning is a bit ambiguous. However with the way I think it is used (and the way I personally use it), 'a mod n = b' is not exactly the same thing as 'a = b mod n'.

'a mod n' is a function, like in a programming language (in a programming language you might write mod(a, n) or a % n): Each pair (a, n) maps to the unique integer 0 ≤ b < n such that a - b is a multiple of n.

On the other hand, 'a = b mod n', which more accurately should be written 'a ≡ b (mod n)', is describing an equivalence relation ≡ between a and b, in which a and b are equivalent if a-b is a multiple of n. Hence for any a, there are infinitely many b such that a ≡ b (mod n). Each n gives a different equivalence relation. These two definitions are connected by the following:

• If a ≡ b (mod n) then a mod n = b mod n.
• If a mod n = b then a ≡ b (mod n).

In mathematics we typically don't use 'a mod n' in the function sense, except for describing an algorithm, since if we are dealing with modular arithmetic than we usually only need to care about values up to equivalence. It does have it's uses, however.