Use Case is prepared by business analyst depending on the 
customer requirements, where as test case is written by 
testers depending on the BRS,SRS,FRS and use case documents.

A use case is a methodology used in system analysis to 
identify, clarify, and organize system requirements. Use 
case gives the idea of how to use the application and what 
is the business functionality.

By using use case we can derive multiple test 
cases/scenarios for single use case.

Ex: For login screen login into the application by using 
username and password is a use case.

Possible ways the user should and should not login into the 
system is a test case

use case is the the single interaction of system and actor
while test case is the combination of all valid and invalid
inputs and out puts

for more details visit

www.s-qa.blogspot.com