Using a network dataset and a Route analysis layer, you can load in multiple points and on solve, the stops attribute table contains a new column with a name that starts with "Cumul_". For example, if you solved on an impedance attribute called Meters, this column will be named Cumul_Meters. It contains the cumulative meters from the start to reach the individual points. The difference between any two will be the difference between them.
So you could select only those points with the same attribute, load them into Stops of a route layer, solve and use the Stops attribute table as needed.
Jay Sandhu