You can use it e.g. for a range request.
example: your client requests the resource http://examp.le/foo?id=3 and the Contents-length is 4096 but your client only requests the first 1024 bytes. It can then (at a later time) request the remaining 3072 bytes but that doesn’t make sense if the resource has changed meanwhile.
edit: Also you might not want to change/update data if the resource has changed meanwhile. E.g. you request a customer record and edit something. If someone else has changed the record in the meantime this might lead to inconsistencies. Therefore send your updates with an if-unmodified-since(-I-retrieved-the-data) header and the webserver will/should reject your updates if the record has already been changed – your client can then request the “conflicting” data.
edit2: since you’ve asked for “any practical use of If-Unmodified-Since in the wild”:
see http://msdn.microsoft.com/en-us/library/dd179371.aspx#Subheading1.
Let’s assume you’ve first requested the Blob properties. Now you know e.g. the Content-type and Content-length (maybe you need this for some kind of allocation). Someone/something might change the blob before you send the second, Get Blob request. If you send the value of Last-Modified as value of the If-Unmodified-Since header the server will respond with the appropriate error code if the blob has changed.
Those are examples of an optimistic lock/stamped lock as a means of concurrency control, where the value of the Last-Modified header serves as the guard token. See e.g. https://en.wikipedia.org/wiki/Optimistic_concurrency_control