What are the limitations of commercial-grade DNA genotyping compared to full sequencing?
23andme briefly describes the technology they use here. They are testing the genotype of your DNA at roughly 1 million locations. The technology they use to do this is known as a microarray.

The limitations of using a microarray, as compared to sequencing, is that you will only find what you are looking for -- people often describe the disadvantages of microarrays as compared to sequencing with the streetlight effect/metaphor.

Arrays can only measure regions of the genome that they were designed to probe. Technically, if they probe 1,000,000 million locations, and the human genome is roughly 3.4 billion bases ... you can do the math.

In practice, SNPs tell you a bit more about just the nucleotide being interrogated, due to linkage disequilibrium (cf. tag/proxy SNPs), so the array might tell you more than you expect.

Of course, modulo sequencing errors, whole genome sequencing will tell you "everything" as far as recovering the NTs that make up your genome, but how much information that will provide you is another thing altogether (for now, that is).