Anhydrous magnesium sulfate to replace epsom salt?
Yes, they become the same, with some rare exceptions. There is usually difference in the dissolving kinetics, but the final result is the same. 

Sometimes there are different thermal effects, Hydrates have sometimes tendency to cool the solution, while anhydrates to warm it. It is caused by released hydration energy of the anhydrate.The typical example is calcium chloride.

The obvious thing is to recalculate the needed amount for the contained water.