I am an applied physicist - my PhD is in Applied Physics.
Applied physics is usually considered a bridge between physics and engineering. Our research tends to focus on practical applications of existing physics knowledge, with less emphasis on fundamental research on how things work. The education that I received was comparable to that of other physicists, but included more engineering topics - in my case mostly electrical engineering because my specialty is laser physics, and the design and construction of lasers, and many other parts of practical optics are taught in the engineering schools. I have colleagues who graduated from the same program but whose concentrations are in biomedical engineering, MEMS devices, x-ray tomography, and a variety of other areas. Most of them work in industry, though some work in government labs or teach.
So how does an applied physicist differ from an engineer? Well, if both of them have doctoral degrees ... not much, especially if their areas of expertise are similar. When I work on an engineering project my role is usually at the beginning, and there are concerns about what techniques might be appropriate. Sometimes some experimentation is required, or interfacing with the research group, or conducting literature searches.In a recent project the goal was to determine if scrap metal could be sorted by means of ultrasound. I researched what has been done, developed a mathematical model for the proposed process, and carried out simulations of responses that could be expected for the materials of interest at different wavelengths and other conditions. Having established sensitivity criteria I was then able to conduct some experiments to evaluate the different conditions in a more "real world" setting. This process took about 50 hours over a period of one month - this was a part-time project. At the conclusion I prepared a report with a summary and all of the details. If they decide to go ahead most of the work will be regular engineering: real time computer systems design and implementation, electrical and mechanical engineering design for the transponders and fixtures, recommendations for the procurement of items - and when the prototype is built and installed on the test production line - then I will become involved again during the evaluation and testing phase - but only to check the data acquired against the expectations contained in the earlier report.The theoretical physicist will study more mathematics, and focus on the theoretical courses instead of the engineering and applications. They usually work with the experimental physicists (who are much more numerous!), and help with the theoretical predictions for a planned experiment, or with the analysis of data after the experiment. Some are also specialists in computer simulations.
But many experimentalists do their own theory and analysis; it depends upon what is being worked on, and where the major issues lie. As you can see from the ultrasound scrap metal sorting project, I often do my own theory. But these are not fundamental research!
I've also worked with many engineers over the years, and some of them can do each of the things mentioned above. The main difference is that engineers usually work on the implementation of known technology - perhaps building more efficient or less expensive electrical generators, or a robot that can pick up a cup and fill it with hot coffee without spilling, or an automotive tire which contains tread sensors.
A lot of engineering is focused on the integration of systems while following established protocols for safety and efficiency - while meeting budget and schedule targets. While many of these same constraints exist for physicists, especially safety and budget, it is usually difficult to do research on a definite schedule. Research tends to be incremental.
Different people may slice this a bit differently; here is the program description for the University of Michigan Applied Physics program: http://www-applied.physics.lsa.umich.edu/