People have already adequately covered the difference between an EE and an electrician (and a EET or electronics tech) but I’m compelled to offer an additional perspective from across the aisle in the Mechanical Engineering section.
Most people who aren’t familiar with mechanical engineering think that we all design cars. “Hey, can you fix my engine.” Er, yeah, okay…looks like a blown bearing transformer to me; maybe you should take that down to Pep Boys and…uh…have the serpintine shaft replaces. 
Seriously, while I’m a fair shade tree mechanic, at least as far as Volvos and Subarus go, the only class in university I had that even touched on the mechanical workings of internal combusion engines was a lab in which we had to measure the brake horsepower out of a turbocharged diesel. We had some thermodynamic theory on the Otto and Diesel cycle, and some general coverage in cam design (but not specifically for engines), but that’s the extent of what most engineers know about your car engine.
In the “Senior Survey” class (a lecture hall seminar which mostly consisted of various alumni coming to lecture us on what the real world was like) we had one speaker who asked the class how many had ever changed their own oil. A few hands went up–distressingly few, actually–so then he asked how many had ever opened the car hood and looked at the engine. A few more hands went up, but the sum total was that less than a third of the class had actually even looked at an engine.
Seriously.
Whoa. I mean, I’d fallen into mech. eng. after drifting through physics, mathematics, computers science, and electrical engineering, so it wasn’t exactly my life’s dream, but heck, at least I’d turned a few bolts in my time. I wouldn’t trust most of these kids with a phillips head screwdriver. A couple years after my freshman year the School of Engineering started having incoming students take an “Intro to Engineering” class where they had to disassemble electrical and IC motors, air conditioners, and so forth, so hopefully future batches of engineers will be better acquainted with the internal workings of ordinary devices than were my compatriots, but if someone tells you that they are a mechanical engineer, I wouldn’t go inviting them to change a tire, much less operating on an ailing powertrain.
As for EEs, there are a few I’ve met that I doubt could even put together a simple band-pass filter, much less design a complex circuit. The field covers a lot of ground beyond basic circuit design, into power generation, VLSI, semi-conductor materials science, and even applied quantum mechanics. VCR repair, however, is probably not on the list of undergraduate courses.
Stranger