Genetics, broadly, is the study of the processes of heredity, how genes are passed on; while genomics is more the specific genes and genetic code associated with a specific organism or species.
The difference between clinical genomics and personal genomics is rather like the difference between apple harvesting and cherrypicking.With apple harvesting, you put a blanket under a tree while someone climbs up and shakes the branches to try to get as many apples to fall as possible. You gather all the apples from one tree before moving along to the next. With clinical genomics, the goal is to do a deep and complete examination of the genes likely to be associated with a clinical problem, usually looking at the full scan of all the code of a specific gene known to relate to the clinical concern.
With cherry-picking, you pick the cherries within easy reach, the ones that are ripe right now, and then leave the rest to ripen later or for the birds to eat. With direct-to-consumer personal genomics services, they examine just the few bits of code with strong associations to a variety of possible clinical concerns, whether or not the person has any signs of problems. Personal genomics services give clues, not answers.
Here are some resources to learn more.
Genetics Home Reference: What is a genome?
NHGRI: A Brief Guide to Genomics
The Jackson Laboratory: Personalized medicine and you: Genetics vs. Genomics
World Health Organization: WHO definitions of genetics and genomics