The basic of genetics is you get one half of your moms DNA and one have of your dads. Many things only express themselves (we can tell you have the gene from the outside) if a person is homozygous for it. What that means is both parents give that same gene. If you are heterozygous, only from one parent, it may not be a problem. For instance, I think Cystic Fibrosis works this way. You can have two parents who are healthy, but if they both are heterzygous for CF and give that half of their DNA to the new embryo, and the embryo is homozygous, it will be sick. It is a recessive trait that only makes someone sick if there is not a domininant trait, normal gene, to balance it. The parents may have no idea they carry the gene, unless they are tested. Red hair is recessive. If both parents have red hair, all children will have red hair, because both parents have the recessive trait red on both sides of the DNA. If one patent had brown hair, but recessive for red, there is a chance of children with red or brown, if there is a paremt who is brown, brown, from both of his parents, the child won’t get her mommies bright red hair, because the father will only giving the dominant brown to the baby.
Ok, so back to your question. Just because children look like each other, does not mean other genetic traits are perfectly similar in both children. Red hair is a different part of the genetic code than susceptinility to breast cancer for instance. Although, there is a theory, maybe it is a fact now, that certain parts of the gene strand might move together as genes move in the beginning process of splitting to form an egg or sperm, meaning they might see more breast cancer in red heads (that is completely made up as an example).
Some ethnic backgrounds carry certain genetic diseases more often, so if part of the reason the parents and children and maybe the whole extended family look similar, it could be they come from the same part of the world, and so carry very similar traits and genetic diseases. Ashkenazi Jews get tay sachs more often, people from the Medeterranian we see thalessemia and intrinsic difficiencies, etc.
To sum up people who both have recessive traits for bad diseases, may have no idea they carry the disease, family members have a higher chance of having similar genes, and so when they make a baby, like a brother and sister, or first cousin, they are more likely to have a baby that gets the bad thing from both parents and actually be sick from it. But statistically it is actualky only a slightly higher chance than the averge population, it is not how the public perceives it that you have a huge chance of being very sick. I think people think it is more likely the baby will have some major malformity or illness, but it is still more likely the baby will be fine.