It's kinda splitting hairs to define the two separately. The terminology around Sensory Integration has changed and therapists will often use different phrases for similar issues.
As best I can understand it the difference is this:
Sensory Integration is how our brains process incoming sensory information, and put it into the 'mix' with cognitive information, language, goals, priorities etc.
So, you can simultaneously drive, and listen to the radio. As your NT brain can cope with that level of sensory info coming in.
But if the level of demands is increased (say you have to drive in mid-London traffic on a wet, dark November Friday evening, with a crying child in the back, while trying to get to an address you dont know) then you will reduce any other sensory demands (or you will melt down like me. ). Melting down in that situation is normal.
Melting down cos you have to get dressed in the morning is not seen as normal. But if your child cant cope with the sensory demands, it's the equivalent of the driving example to him
Sensory modulation is how we respond to the information. See here.
So, one child will respond to hitting their head on a cupboard door by ignoring it (maladaptive and under responsive, will repeatedly make the same mistake), another will say ow and rub their head (adaptive response, will learn not to do that again) and another will scream blue murder (maladaptive, over responsive, may avoid all cupboard doors forever after).
Two of these dont have the ability to 'modulate' IYSWIM