As pure adjectives, I can see no difference. If you are talking about whether a particular organism is alive or dead, alive and living are exact synonyms. For example: "Never cook and eat a lobster that is not alive/living when you cook it. You will only forget this once!"
Alive is pretty strictly a biological adjective. There are some exceptions, but for the most part alive is simply the opposite of dead.
Living, on the other hand, has many uses based on its being the present participle of to live.
You can be living with someone, i.e. cohabiting as unmarried sexual partners.
You can be living the good life (a practice I highly recommend).
You can be living with cancer, i.e. you have it but it hasn't killed you yet.