What exactly do electronic engineers do?
Electrical engineers in theory design circuits.  In practise, a lot of other stuff comes with that.  The amount and content of the other stuff will vary between specific EE jobs.  However, one common theme will be designing and writing firmware.  Firmware in small resource-limited processors that interacts directly with hardware is the domain of EEs, not software engineers.  The higher level the software it is, the more it's a software engineer problem and less a EE problem.

Designing a circuit will include the architecture (higher level concept) and the detailed design (what specific parts to use, resitor values, etc).  Depending on the company, the EE may then get envolved with layout and board design or not.  I know some EEs that feel board design is someone else's job, and there is at least one person I know in this area that does just board design from schematic as a consultant to service those cases.  I personally consider board desing integral to the overal electrical design and prefer to do it myself.

Electrician is a very different profession from a electrical engineer.  A electrician is a tradesman.  He wires houses and the like, and has to know the electrical code.  In fact, he will have taken a test and received a license and can legally sign off on electrical wiring installations proclaiming them to be to code, whereas most EEs don't have that license.  The electrician doesn't need a college degree, but a EE does.  The electrician has no reason to understand how electronic components like transistors and even capacitors and resistors work other than how to install them an hook them up to the extent this comes up in power wiring installations.

Those that suck at math need not apply to any type of engineering, and EE is no exception.