Why the smell of sweat greatly varies across the body?
Sweat is actually odorless regardless of what type of sweat gland it comes from, of which there are two types - apocrine and eccrine.

As suggested by canadianer, the odor comes from the mixture of bacteria and the resulting sweat. With regards to why your bacterial flora is so diverse is likely due to the different micro-environments found around your body.

In your armpits and feet (arguably where the strongest odor subsides), you'll find more anaerobic bacteria - bacteria that prefers low oxygen environments and sometimes, higher temperature, humid environments. Because they don't use oxygen to metabolize, their by-products in making energy are often organic acids, commonly lactic acid. Some anaerobic bacterias even have by-products containing sulfur. 

Meanwhile, in less warm and damp environments on your body (i.e. face, hands), aerobic bacteria dominate. Oxygen is abundant and the temperature is just right for those strains of bacteria. Their by-products are much less putrid

EDIT: In addition, apocrine sweat is also high in the nutrients, such as lipids and proteins that bacteria can feed on. Once they metabolize the stuff in apocrine sweat, they start releasing the aforementioned by-products.