According to the central dogma of molecular biology, your body takes a piece of DNA and transcribes it to mRNA in an irreversible chemical reaction. Your body makes mRNA segments all the time, and if this reaction was not irreversible, then your body would keep on transcribing DNA into mRNA and reabsorbing it. Your genome would naturally get so messed up that you'd die as a single fertilized egg.

A virus manipulates your body into translating its entire genome into mRNA and then into proteins. mRNA vaccines do not give you the entire genome, but rather a segment of already-transcribed mRNA -- again, which cannot be re-inserted into DNA -- which then gets translated into a protein. This protein is recognized as foreign and stimulates an immune response. This is safer than every gene in a virus being activated inside of you, it's just one gene, that cannot do anything on its own, other than impart immunity once it is seen.