Is it normal for brakes to feel spongy or soft to push on a 2003 lexus es 300

I replaced my brake pads but my brake pedal
still feels soft and spongy when pushing down
on them. The car brakes and reacts when I
push the brake pedal down but it is not stiff.  
When my car is turned off my brake pedal gets
stiff after pumping it but when I turn on the car
the brake pedal goes down and feels spongy
again. So I took it to a mechanic and he
replaced the master cylinder on it, but the
brakes feel the same.  The mechanic said it
was normal and that brakes are not supposed
to feel stiff.  Is this true? Ivery driven other cars
with stiff brake pedals but he said that cars
have different abs systems on them and that if
my pedals were stiff they wouldn't work.