Finding differences between shapefiles using ArcGIS for Desktop?

With the Compare Feature Tool, you should just choose your Segment_ID field as the sort field in the dialog.

The [sort] field or fields [are] used to sort records in the Input Base Table and the Input Test Table. The records are sorted in ascending order. Sorting by a common field in both the Input Base Features and the Input Test Features ensures that you are comparing the same row from each input dataset.

If you are wanting to get the Segment_ID field as part of your output table, try joining the output Compare Feature table to your source Test Table on cf.Object_ID=tt.Object_ID, then export out the joined table with your Segment_ID.

You could easily put this into a model for easier repeatability.