Greedy search (for most of this answer, think of greedy best-first search when I say greedy search) is an informed search algorithm, which means the function that is evaluated to choose which node to expand has the form of f(n) = h(n), where h is the heuristic function for a given node n that returns the estimated value from this node n to a goal state. If you're trying to travel to a place, one example of a heuristic function is one that returns the estimated distance from node n to your destination.
Uniform-cost search, on the other hand, is an uninformed search algorithm, also known as a blind search strategy. This means that the value of the function f for a given node n, f(n), for uninformed search algorithms, takes into consideration g(n), the total action cost from the root node to the node n, that is, the path cost. It doesn't have any information about the problem apart from the problem description, so that's all it can know. You don't have any information that can help you decide how close one node is to a goal state, only to the root node. You can watch the nodes expanding here (Animation of the Uniform Cost Algorithm) and see how the cost from node n to the root is used to choose which nodes to expand.
Greedy search, just like any greedy algorithm, takes locally optimal solutions and uses a function that returns an estimated value from a given node n to the goal state. You can watch the nodes expanding here (Greedy Best First Search | Quick Explanation with Visualization) and see how the return of the heuristic function from node n to the goal state is used to choose which nodes to expand.
By the way, sometimes, the path chosen by greedy search is not a global optimum. In the example in the video, for example, node A is never expanded because there are always nodes with smaller values of h(n). But what if A has such a high value, and the values for the next nodes are very small and therefore a global optimum? That can happen. A bad heuristic function can cause this. Getting stuck in a loop is also possible. A*, which is also a greedy search algorithm, fixes this by making use of both the path cost (which implies knowing nodes already visited) and a heuristic function, that is, f(n) = g(n) + h(n).
It's possible that to this point, it's still not clear to you HOW uniform-cost knows there is another path that looks better locally but not globally. It should become clear after telling you that if all paths have the same cost, uniform cost search is the same thing as the breadth-first search (BFS). It would expand all nodes just like BFS.
