As a native English speaker, I have never used the word "consequent" before. "Consequence" is probably the only word you should worry about, but they do mean close to the same thing.

To follow up on the answers below: consequent can be an adjective or noun, but it is a word we don't use (I never have); consequence is a noun and is used commonly; consequently is also very common - it means "as a result".

Both of the words have same meaning but most commonly we use "consequence". Consequence is used as a noun while consequent is used as an adjective.