What is the difference between having a crush on someone and loving someone? What is the psychological basis of this?
Adding to Jeromy's answer, where he provides definition of love I am addressing the nature of the crush vs a more adult love style.
By definition, the crush is mostly experienced within teenagers, this group of individuals has less impulse control and ability to moderate their behaviour and feelings due to brain development and hormonal changes. These physiological conditions combined with the newness of sexual feelings and relationships, makes the individual less capable of managing their emotions.
The teenage brain has more white matter and less developed grey matter, meaning the higher functions of reason, moderating and inhibiting extremes of behaviour and emotion are not as well developed as in a mature adult brain.
Meaning they are rules by their limbic system. So sexual attraction or admiration would be felt intensely.
In contrast to the adult who has a more developed brain, stable hormone levels; and, usually, has experienced the teenage crush, and had adult relationships.
The Urban Dictionary
The Free Dictionary
This wikipedia on the Biological basis of love explores how the state that we define as love is also based within the limbic system.
This question Studies linking brain chemistry to sexual infidelity explores the biochemical aspects of sexual attraction and fidelity.
That teenage feeling
You may find Sternberg's triangular theory of love as useful framework for distinguishing different types of love. Quoting wikipedia, the theory describes types of love in terms of three dimensions:
Using this framework, "loving someone" is the general term, and "having a crush" would be a more specific variant. So perhaps a crush could be conceptualised more in terms of passion and less in terms of long term intimacy and commitment.
Alternatively, you could just look at definitional concepts of a crush. For example, the term "crush" is commonly used in contexts where feelings are not reciprocated. I associate it with younger people (or when seen in older adults, a youthful approach to love). A crush often implies a short term quality, such that the feelings may change from one month to the next.