Line sections between points in ArcMap?
How to get shape with lines between points which have to lay on the same line entity?
This can be done using linear referencing.
1) After creating routes from you lines, you can extract the M value of those lines for each point using locate feature along route.
2) With these value, create a table including the start and end M values for each line. (table join) make sure that you use the line ID to avoid using two points from different line entities.
3) Use make route layer event to create the lines based on your table.
IF you have an advanced licence, you can also use another method.
1) Use "split line at points" to split your lines where you have a point.
2) Perform a spatial join with the result, and select by attribute the lines with 2 points. Those will be the lines that join two points on the same line.