Difference between "live life" and "live a life"

Live life might be used in a positive and affirmative sense "live life ...to the full". Live a life could be slightly more negative, and be heard as "get a life", implying that the person needs to do something different, and more positive. Context would be key though, because live a life could be said in an affirming way.

"live life" and "live a life"

In "live life" life is uncountable and refers to that which is experienced whilst living, i.e. "life in general".

"Live a life" is, as Edwin Ashworth points out, wrong because it is incomplete - it requires a complement, i.e. "Live a life of happiness/worthy of your calling/that people will remember, etc."

In "Live a life of happiness", "life is countable as it is one life from many possible types of life, and "a" = an example of a life of happiness.