Why do people of a certain race tend to look similar to each other? I understand why skin color will differ depending on ancestral environment, but I don't see why hair color would have any effect on an individual's chance of survival. Ditto for slight variations in facial bone structure and the various other differences in appearance that define the races.

The simple answer is we do not know but there are a number of possible explanations:-

1.    There were advantages to the changes you highlight we are just not aware of them. Lightening of the skin to improve VitD synthesis is an obvious example as you say.
2.    The genes that code for many of the facial features (transcription factors) do very many other things as well. So the visible changes you see actually represent a completely different survival advantage that might have something to do with the immune or nervous systems.
3.    There are no advantages to the changes – they just represent natural “genetic drift” due to random mutations in a relatively small and isolated population. If they confer no advantage or disadvantage there is no pressure on them to be lost. This is thought to be the basis for the so called "founder effect" (see the wiki link below).

One important thing to note: recent large scale genetic studies have shown far far fewer genetic differences between racial groups than we had previously expected. The concept of "race" especially as determined by skin colour has been abandoned by many geneticists, physical anthropologists, and evolutionary biologists. The term "race" is a social and political construct, and many believe that differences in say facial features or disease prognosis is principally due to behavioral or environmental differences rather than genetic ones.

My own view is largely in agreement with the above - in many cases genetic differences clearly do play a major role in the incidence of a disease or its prognosis but that has very little to do with "racial" inheritance.

The wiki page on this is comprehensive:-

I would guess others on this site will have much to say on this subject!