Take both polylines and run the intersect tool. You can change the Output Geometry to be a point. See image below:
This will create a point feature where there are overlaps between the two lines. Using the "Split Lines at Point" Tool you can cut the streamlines at the road crossings. You can find the split lines tool under "Data Management -> Features"
The tool is only available in an Advanced License.
A work-around for that would be to use ET Geowizards (Link to site) or you could use the "Split Lines with at Points" tool in the Q-GIS SAGA toolbox. Looking through the SAGA Toolbox I do see a split line with line tool...so that may also be a solution to look into if you aren't limited to purely ESRI Software.