Grinding poverty is longlasting poverty that seems to have no end. Abject poverty is extreme poverty, the worst that one can imagine. Grinding, abject poverty is the extreme poverty that many people in the world experience from the moment they are born until they die.

The images are really quite different, even though they both refer to being very poor. "Grinding" means... well, the poverty crushes you. You can't escape. "Abject" means you have been 'thrown down' into poverty.