While trees are a type of graph, graphs can have more functionality than trees. For example, recall that to be a single tree, there could be no disconnected pieces.
Connectedness: Graphs do not require being fully connected. There can be disconnected portions within a graph. For example, the following graph shows all of the students in a sophomore biology class. There is an edge between two student nodes if they are Facebook friends.
Graphs can also have loops. In a tree, this would be like a node being its own parent, which is not an allowable condition.
Loops: Loops are edges which connect a node to itself. These can be useful in depicting graphs that show control flow in programming. In this example, node A is connected to node B and node A is connected to itself.