Each of my cells has, at its heart, a nucleus containing a complete set of....

All cells in your body contain the same DNA, yet they look different. The reason for that is due to epigenetics. Epigenetics is the idea that certain genes get switched on or off depending on the tissue. Switching the genes on or off is done in a variety of ways, through methylation of the DNA or acetylation of the histone proteins, preventing certain genes from being transcribed. This is called differentiation, and not too many cells in an adult human are pluripotent. You find undifferentiated (stem cells) during the early stages of development.

There are two kinds of methylation that are relevant here: DNA methylation, which can occur at sites where a cytosine is followed by a guanidine (CpG sites), and histone methylation. To my knowledge DNA methylation is used exclusively for repression of nearby genes. Histones are the proteinaceous spindles that DNA is wrapped around in the nucleus. Depending on what amino acid is methylated, histone methylation can cause activation or repression of transcription for nearby genes.

When a cell becomes committed to a particular developmental program, it permanently represses some of the genes that it will not need to use using DNA and histone methylation. This ensures that a cell (almost) never adopts two fates by mistake. It may also permanently turn on the genes it will need through histone methylation and acetylation. This is why methylation is important to cell differentiation.

You might think that totipotent cells would have less methylation than multipotent cells for that reason (and this is probably the answer your teacher wants). But it turns out that in pluripotent stem cells, histones are often "bivalently" methylated. That is, they are methylated in two places on the same nucleosome: one methyl group that normally causes transcriptional repression, and one that normally causes transcriptional activation. When the cell differentiates, one of these methyl groups is clipped off and the remaining one determines how the histone will regulate gene expression. So in this sense, totipotent stem cells have more methylation.