Why is there a difference between life-insurance and general (non-life) insurance mathematics? I know that actuarial mathematicians specialize in either life or non-life insurance. My question is, what is the difference between these two fields mathematically? Why is there a need for specialization? Why isn't it the same mathematical theory and concepts that underlies both industries?
Because that's what insurance mathematics is, no? You have some claim that occurs randomly, and you need to model that randomness, and you need to risk-manage it. 
So why is the mathematical theory split up in two? How is it split up? What math does a life-insurance actuary need that a non-life actuary doesn't? 
 A: I work in non-life insurance and know a number of people who work in life insurance.  It amounts to this: Most of the interesting work in non-life insurance is around statistical inference.  You have very heterogeneous populations of risk that change over time.  (Think of how different buildings are from one another and how safety features are constantly improving.)  You have to make statistical inferences from limited historical data to draw the best conclusions you can about, for example, the cost of risk of loss by fire to a certain building and its contents. And one also makes inferences about the risks of security class actions, of medical malpractice claims, of ships being attacked going through the strait of Hormuz, etc.
In life insurance, there is only one insurance risk you are evaluating...mortality.  And it isn't going to be suddenly changed overnight by a social change like tort reform or by adding fire-resistive cladding to people's skin.  So actuarial organization produce standard tables for various countries and actuaries do studies of how much they need to alter those estimates based on health status, etc.  (But typically they don't sell insurance to someone whose health is really terrible, so they don't usually have to grapple with the fast-changing items like the life expectancy of someone who was just diagnosed with stage IV cancer.)  
On the other hand, because life insurance needs to be a long-term product to provide financial protection, life insurers embed features in their products that provide an investment return on the value that accumulates in the policy because the annual premium is level whereas the risk of death is extremely small at first and rises later.  That is often a fixed interest rate, but it is also often based on stock market returns and is complex enough that many life insurance and many annuity products (which life insurers also tend to sell) can only be distributed by licensed securities brokers.  And even a policy that provides a fixed interest rate needs to be carefully hedged over long periods of time.  The result of all this is that life insurers have less (not none, just less) need for sophisticated statistics and an extreme need for sophisticated finance to manage the investment assets that hedge their risk, i.e. that will allow them to provide the promised returns to their customers regardless of investment market conditions.
To sum up, non-life insurance is mostly about statistics and about the liability side of the balance sheet; life insurance is largely about finance and the asset side of the balance sheet.
