Subsequently means afterwards and says nothing about why: "he resigned; subsequently the share price fell." Consequently means as a result and asserts causation: "he resigned; consequently the share price fell." Choosing the wrong one either claims a cause you cannot support, or hides one you meant to state.
The core difference
Subsequently places two events in order. Consequently claims that the first produced the second. Sequence is not cause, and swapping these words changes what your sentence asserts.
Why the distinction matters
In anything evidential — a report, a case note, an analysis — this is not a stylistic preference. Writing "the drug was administered; consequently the patient recovered" states a causal claim that "subsequently" would not. If the evidence only shows order, subsequently is the honest word.
- subsequently = after that: The policy was announced in May and subsequently withdrawn.
- consequently = therefore: The policy was unworkable; consequently it was withdrawn.
A test
Replace the word with therefore. If the sentence still holds, you mean consequently. If it now claims more than you can show, you mean subsequently.
