These two words are to do with cause and effect (or not), and of Thing B happening after Thing A. But they don’t mean the same, even if people think they do!
Something is consequent when it follows on from something else as a consequence, i.e. as the result or effect of that first thing. “She dyed her brown hair platinum blonde and consequently stood out in the crowd” – so, she stood out from the crowd because she had platinum blonde hair.
Something is subsequent when it merely follows on from something else in time, i.e. not as a result or effect of that first thing. “She had brown hair and subsequently went blonde” – she didn’t go blonde because she had brown hair, it was just the next thing that happened. She could have had black, red or blue hair first; the original hair colour doesn’t affect the outcome.
“She had black hair before she dyed it red, consequently she had to bleach it first before applying the red dye subsequently”. I don’t think anyone would actually ever utter that sentence, but hopefully it explains matters!