Test Case - is an unit which consist of:
1. Object which is under test
2. Input data
3. Actions to be performed upon the object
4. Expected output.
Test Steps - Step-by-Step description of Actions to be performed
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Very close, but the test Steps contain Actions to be performed, Input Data, Object which is under test, (if it is an object), Expected results, a place to indicate Pass/Fail for that step, as well as a place to enter information about anomalys observed and symptoms for failed condition.
Whereas a Test case is a set of steps along with identification of who created it, when it was created, when it was updated, what application & version it was designed to test and what functional area it covers and maybe even what requirement it validates.
Can we summarize this in a such way?
Test Steps - idea how to test
Test Case - entity which represents test steps & include all reqired attributes to get it tracked within the project enverunment