At first glance this seems like sort of a basic question, but these terms get interchanged all the time. If you think it through, this could cause confusion on both the buyer and the equipment provider sides, causing mistakes and mishaps in configuring the correct machine for the job.
I've been involved in the laser business since 1986, presently with Trotec, but previously with different types of laser companies who provide both standard and custom equipment for a wide variety of applications. In that time I have always worked closely with the Applications Technicians who are tasked with determining the feasibility of a project and specifying the correct process AND laser for the job. So this is what I learned from them over the years.
Metal Cutting
Laser Engraving
This is the most common term and simply defined, means that in the process of creating the mark, material is removed or vaporized during the process. An example in our business is CO2 laser engraving plastic for signage. This is often a two-ply plastic with one color called the "cap" laminated to a contrasting color called the "core". The engraving process removes the cap and exposes the core to create a sign that has an attractive look. There is also engraving in the processing of metals. Typically used to guarantee permanence, material is removed to create depth. Usually this is a process reserved for 1,064nm wavelength lasers (YAG for example) as those types of lasers couple well with metals. When engraving metals this way the laser is breaking the machined surface of the object, so oxidation or rusting can occur as a result. This would be a question at the time of feasibility testing. "Do you want depth? If so do you realize that the base material could oxidize? Depending on the answer the next option might be Laser Marking.
Laser Marking
Annealing or precipitation marking is commonly used when processing metal items where the surface of the part must stay intact. Often items like surgical implants, surgical instruments or high precision bearings require this type of marking. With laser marking, the heat of the laser actually redistributes the carbon in the material to create a jet black mark with only micron level surface disruption. If done correctly there will be no oxidation or rusting even under salt spray tests or autoclaving. Some plastics like ABS and Delrin also allow for a nice contrast when processed with a 1,064nm laser. If you processed the same material with a CO2 laser you would get an engraved marked with no contrast. This is all a matter of wavelength and how it reacts to the material OR the pigment in the material.
Laser Cutting
This is sometimes confused with laser engraving but really means totally sectioning a piece or cutting shapes directly through material. Most laser cutting is performed with CO2 lasers as again they interact with the material rather than the pigment within the material. We work a lot with acrylics which can be used for signage, retail displays and many other applications. While cutting, special attention is taken for smooth cuts and what is a called "flame polished edges". There are times where high powered, pulse YAGs are used for metal cutting but even with these materials, high powered CO2's seem to do the best job. In the cutting world, the other technology that is used is water jetting which also does a nice job on metal and stone. Cutting does not always mean thick materials though. Other common applications are paper cutting, leather sectioning and even things like gasket cutting.
So , in conclusion, understanding this basic terminology while you are shopping for a laser or a company to work with will go far to strengthen your communication with the vendor and will give him a better chance of meeting your needs and expectations.