Differntial back up: what happens to existing data?
Since you'd have to apply the differential backup's previous full backup, the end result is identical to if you'd just taken a full backup, made changes, then restored the backup you'd just taken - any changes made would be lost.

There are, however, tools and techniques which would allow you to perform partial restores, so you could limit the application of a differential backup to just objects which had not been modified by your changes.

Since you can't apply a differential to a database that has been recovered ( as in, restored from a full backup without the NO_RECOVERY option or otherwise WITH RECOVERY'd thereafter ) and you can't modify a database that is offline ( in this case, in recovery would be the offline state in question ), trying to do so would not only negate any changes that were made, but require a complete full restore to be applied again before the newly arrived differential could be applied.

More information regarding the reason behind your copy process ( DR redundancy system, development environment, read-only reporting server, etc ) would be needed to make a recommendation on a suitable technique for your specific problem.