In order to understand the difference between a love story and a romance story one must first understand the difference between love and romance.
Love is an emotion where a person feels a strong attachment for another person (or thing). Romance, on the other hand, is the feeling one has in association with love for another. Therefore, emotion and feeling also need to be defined for complete understanding. Emotion is a state of consciousness, whereas a feeling is a sensation. Therefore, a love story tends to be something dramatic and a romance tends to be melodramatic. (Melodramatic meaning exaggerated.)
In Shakespeare's play "As You Like It," a love exists between the couples. It is this love which creates the telling of the romance. While both are very similar, and the question of their similarities is significant, a story can contain love without actually being a story of romance. Given that love is spoken about throughout the play, love is a main focus and the story, in turn, becomes a romance.