NVIC and peripheral interrupts

Hello,
the more I am playing with Cortex-M3 the more I like this solution, but....
I would ask someone more experienced what two ways of handling interrupts are for.
I could do it in NVIC - for instance

IntDisable(INT_TIMER0A);

as well as in pheripheral:

TimerIntDisable(TIMER0_BASE,TIMER_TIMA_TIMEOUT);

I can imagine to disable some interrupt in a pheripheral temporary when something happens - for instance to disable GPIO_FALLING_EDGE when I waited both edges at first and the falling is comming and I measure the time from an edge going first to the second one. Never the less such attempt is rather very rare.
It is probably some reason for implementing interrupts by this way but I am probably not enough smart to understand this.

My life experience taught me not to play with interrupts very much because it's usually a source of serious problems.

Regards

Igor