Love sick means many things. It is the combination of joy and pain felt by the two persons who fall in love with each other. Initially they both hesitate to express their love for each other, but once it is an open secret then they can't live alone and the feeling of wanting to have company of each other desperately increases to such an extent that they can't bear the separation for a single minute. Its a mixed reaction of love, affection, caring and sharing, the fear of separation and what not.
Love sick means the pain that is felt by a person when they fall in love with someone who doesn't reciprocate the same feelings for them. It can also be used to refer to the feeling of desperately wanting to find love.