Great question! I'll share based on my experience as a former systems engineer in Defense/Aerospace. At the fundamental level, Electrical Engineers work with electronic circuitry, the circuit logic, voltages, currents, and electronic components and microchips such as transistors, processors, memory chips, and so forth. Electrical Engineers also work with Radio Frequency energy such as communications systems, antenna design, and transcievers. In addition there is also power engineering, such as electrical power grid, transmission, generation and so forth.
Mechanical engineers typically work on the more physical aspect of all of this. Such as the materials used to build electronic circuits, antennas, components. How do we assembly materials, how do we keep them cool (or warm) enough to operate. How do these system interface with the outside word, such as engines, valves, turbines, pumps and so forth.
Both of these discipline also use Computor Aided Design (CAD) to create drawings and both also use software to solve problems and model systems.
This is only the tip of the iceberg but hopefully helps!
Calvin, that is a great question. There are many career paths for both ME and EE. Since I have been in product development for my entire career, I can only answer based on the different challenges within the product development space. ME design challenges have to consider aspects of product requirements, market needs, ID input, Design for Assembly, design for Manufacturing of the entire product. We work with EE to give them requirements for PCB size, shape, and some component placement. ME also work with EEs on solving EMI, EMC, and ESD issues. MEs work with a lot of different components and manufacturing processes.
From an ME perspective, EE work with circuit design to fit with the ME design that fully meet the performance requirements. The work similarly to the MEs, but their end focus is on the PCB, PCB layout, meeting EMI and EMC requirement. Their job is challenging in very different ways.
As a general rule, the MEs in design spend more time working with the suppliers on site. Both areas are challenging, but the challenges are very different. There are areas in engineering that will allow you to do both.
I think this is not a very clear answer, but I hope it helps.