I know I've read that depression affects people in many different ways, but how exactly does it affect someone physically? Does depression manifest itself as soreness or exhaustion, or do symptoms vary from person to person?
The relationship between "depression" and somatic symptoms is not a linear one. An easy answer would be to say that some people get tired when they're depressed, and sleep all day ("atypical depression") while others have insomnia, etc.
But if you ask the question the other way around, and speculate about the types of mood symptoms that most people with back pain, stomach pain or dizziness have, you get a different answer.
For example, people who present to ERs with chest pain unrelated to the heart are most often diagnosed with panic disorder. Alternatively, individuals who present with lower back pain (in the absence of physical pathology) screen for depression. In the same manner, unexplained dizziness often leads to a diagnosis of anxiety.
You may (legitimately) ask what is the difference between depression and anxiety, especially to a person with pain. The relevance is twofold: different medications are often used for different conditions (benzos for anxiety, for example); and our bodies seem to manifest emotional symptoms in specialized ways, rather than in random physical ailments. For example, if someone has been depressed for ten years, but arrives in the hospital complaining of chest pain that doesn't come from the heart, check again before you blame it on depression.
Read Christopher Lukas's personal journey coping with somatic symptoms of depression: Physical Symptoms of a Psychological Condition.