Eligibility: some scholarships are awardable only to undergraduates and never to graduates, or vice versa.
As a graduate you'll have access to more scholarships and awards, and lose access to others.
Also note that most graduate funding is merit-based rather than identity-based. Thus, it will focus more on academic achievements (published papers, presentations, GPA) etc. rather than being a Welsh American or something like that. (There are some external scholarships for minorities but far more of the funding is not based on this).
Specificity: at the undergraduate level, many scholarships are particular to a programme or degree, but can be awarded to anybody studying that programme or degree.
For example, a science scholarship could be awarded in chemistry, physics, biology, maths, geology, astronomy, and so on. An arts scholarship could be for literature, sculpture, photography, art history, or other areas.
Graduate scholarships are often more focused, not just on a programme or degree, but a particular field. This gets more particular at higher levels where scholarships almost morph into small grants.
For example, instead of being for sciences, it may be for science research related to a specific industry. Instead of being for art, it may be for work done in or about a particular style and technique.
This answer is a community wiki. Please add additional points of difference and a succinct description of them, as above.
