"But if you write code that doesn't change the existing system, you simply test the additions" boils down to the open/closed principle. It's much safer to create a new class (often by inheritance) than to modify the guts of an existing class that already has dependencies.
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Inheritance isn't the only way to re-use code. Using inheritance won't solve your problems, and it tends to give you new ones: http://en.wikipedia.org/wiki/Fragile_base_class
"But if you write code that doesn't change the existing system, you simply test the additions" boils down to the open/closed principle. It's much safer to create a new class (often by inheritance) than to modify the guts of an existing class that already has dependencies.