Once you are out in the workforce, the difference can be very small indeed.
Before I finished college, I was supervisor of metallization for DMOS III (Dallas Metal Oxide Semiconductor) at Texas Instruments. A very hardware and electronics-oriented position, even though I was studying to be a computer scientist.
After I graduated, I started my career as a junior software engineer and then software engineer also at Texas Instruments. There, software engineers and electrical engineers were equal in pay grades and responsibilities.
In Advanced Weapons Programs, we all worked together. Sometimes software engineers were involved in hardware development, but more often, electrical engineers were doing software development.
My direct supervisor at one time was a Professor (Emeritus) of Electrical Engineering at Texas A&M. Dr. P was leading a team of software engineers. In fact, all my supervisors at TI were electrical engineers, with the exception of one physicist (before I graduated from college).