Consequently and subsequently are useful transition words, but they are not synonyms. The word consequently means approximately ‘resulting from’, and is rather like the word ‘therefore’. The word subsequently means ‘after that’, and should be used only to describe the order in which two things happen or occur.
The company announced record profits. Consequently, all employees got bonuses. (Employees got bonuses because of the profits.)
The company announced record profits. Subsequently, the CEO accepted a job at a different company. (The CEO left after the record profits were announced; the profits and the CEO leaving are not necessarily related events.)