Cancer is generally a cell that can replicate uncontrollably on its own (most cells would die if extracted from their very particular environment). Your body has trillions of cells. Many have mistakes, errors and other issues. Most of the trillions of times the errors are very minor and in no way cause uncontrollable cancer - merely a less efficient cell. Only a countable number of times in trillions of cells dividing every day for 80 years are those errors precisely of the kind that would permit uncontrollable replication - cancer.

Aging is different yet. Copying of dna has a Turing issue. It's really hard to copy the end of a Turing tape when you have to hold that tape to copy it. So dna keeps roughly 60 replications worth of dead space at the end of each strand of dna. So even if the replication machinery drops a page or two at the back of the book full of blueprints, it doesn't matter so much. But after 60 divisions or so (a human's lifespan), the replication starts eating into the actual useful blueprints - dramatically increasing the chance of failing to copy an essential gene. These blank pages are called 'telomeres' and the Nobel prize was recently awarded for their discovery.

Aging is a lot of things, but it's not generalized cellular breakdown. Aging is more associated with the gradual failure of larger systems and frameworks in the body in ways that evolution has had no incentive to effect a repair strategy for.

i.e. wrinkles are associated with the loosening of the collagen support framework for the skin. This isn't cellular breakdown - it's just a system which there was never any point evolving a repair mechanism for beyond childhood.