What is the difference between computer engineer and computer programmer?

College programs for computer science or computer engineering, and electrical engineering generally start out with lots of math and physics and other foundational studies. You will also learn about logic systems and how "basic logic gates" are combined to make complex systems. Almost all engineering programs include courses of computer programming so you can be certain you will get a fair about of that. If you specialize in computer science you will learn about things like data structures, algorithms and other foundation material in computer programming. The you could branch out into operating systems, compiler design, database design, computer graphics... the list goes on and on.

If you specialize in hardware you will learn a great deal of physics to help with understanding the basic principals at work inside a microchip. You will learn about basic circuits (and in some schools how to build them), what they do and how to combine them into more complex systems. If you are really interested on hardware you can take a look at FPGA design. This is what modern hardware developers use when they need to build something huge like a game console or a graphics card. FPGAs are cheap and fun to play with because you get to use software to make hardware all at the press of a button. It can be a lot of fun and very rewarding. I highly recommend you google both computer science courses and computer hardware courses and see what interests you the most.

Also, MIT offers courses online so you can learn some of these things online for free! (Google MIT courseware)

Technically speaking, a computer programmer builds and maintains software programs. A computer engineer is someone who is trained in both hardware and software, and could develop a complete system using either of those two. Computer engineering marries hardware roles (e.g., electrical engineering) with software roles (e.g., programming). If you're interested in working with computers, I'd suggest you get started by playing with both hardware and software to figure out what it feels like on each side. For hardware, pick up an issue of Make Magazine (or check out their website) and find some beginner's hardware project to try out. For software, try learning javascript using codecademy.

One very simple difference to understand is computer programmers focus on software and computer engineers focus more on hardware.

If you are looking to build the next great gadget, you want to be a computer engineer since building a gadget needs hardware and software knowledge that computer engineering gives you. Instead, if you want to build the next great app or popular website, you would be better served acquiring skills of a computer programmer.