Though you may hear both terms in reference to obtaining information about DNA, genotyping and sequencing refer to slightly different things.
Genotyping is the process of determining which genetic variants an individual possesses. Genotyping can be performed through a variety of different methods, depending on the variants of interest and resources available. For looking at many different variants at once, especially common variants, genotyping chips or arrays are an efficient and accurate option. These do, however, require prior knowledge of the variants you want to analyze.
Sequencing is a method used to determine the exact sequence of a certain length of DNA. You can sequence a short piece, the whole genome, or parts of the genome (such as the “exome," which are the parts of the genome that contain genes). Depending on the location, a given stretch may include some DNA that varies between individuals, like SNPs, in addition to regions that are constant. Thus, sequencing can be used to genotype someone for known variants, as well as identify variants that may be unique to that person.
The analysis performed by 23andMe is genotyping, not sequencing. Sequencing technology has not yet progressed to the point where it is feasible to sequence an entire person’s genome quickly and cheaply. It took the Human Genome Project over 10 years involving work by multiple labs to sequence the whole genomes of just a few individuals.
For those with specific needs (undiagnosed medical issues, for example, or intense curiosity and a generous budget), sequencing will become a more and more attractive option. For now, though, genotyping technologies such as those used by 23andMe provide an efficient and cost-effective way of obtaining more than enough genetic information for scientists - and you - to study.