A User Story when decomposed to tasks that are individually assignable to developers may or may not be more granular and constrained in scope than a Use Case Scenario. A User Story is about the need of the User -- a Goal or Outcome from their using the System.
Examples of User Stories are:
- I am a Customer and I want to pay my account balance online --a pretty high-level view
- I am a Customer and I want to update the expiration year on my stored credit information -- a pretty granular view.
At the highest level of abstraction a Use Case sounds very similar -- Customer Updates Card Expiration Year -- but here its a statement of function rather than goal.
As the granularity of the Use Case scenarios are defined they become more about function and procedure.
The Post-condition of a Use Case Main Scenario should be the same outcome as that stated in the User Story -- Customer's Credit Card Expiration Year is Updated.
Use Case Scenarios describe step-by-step either in text or process flow chart (not necessarily UML or BPM - I use a standard cross-functional flow diagram for clarity and ease of use by untrained consumers of the use case.
The bottomline is that User Stories describe Needs and Outcomes (What the System Needs to Deliver) whereas Use Cases describe interactions between Actors required to meet the goal - AND what can go wrong (Extension, Alternate or Error Scenarios) (how the user interacting with the System achieves that outcome)
For a discussionin depth on this topic I suggest reading http://c2.com/cgi/wiki?UserStoryAndUseCaseComparison on Alistair Cockburn's website. Since he's a signatory of the Agile Manifesto, the person who coined User Stories and has been considered a Use Case expert for the last couple of decades I think he's an excellent source for more info.