to use the adverb "really" you are expressing the urgency and importance of a certain feeling or state of being. Examples: Jane really needs to find a job, or she will have to move out of her apartment. His meal was really delicious, no wonder the judges gave him first place. I was really tired after walking around New York all day.
"Very" is an adverb that adds some emphasis to the verb or adjective that it is paired with; we use "very" to let people know that something is above an average rating; that it is very good, as opposed to plain old average good. Examples: The teacher was very angry with Jason, you could see his face turn red. Pulp fiction is a very original movie. In Canada, the winters are very cold, so you better be used to freezing your butt off.