I think the quote is ambiguous. It could be taken as omniscient conscious awareness of everything throughout spacetime, but it could also be taken as a more limited awareness concerning any specific event in one's spacetime vicinity--seeing not only what actually happens at that event (what 'is'), but also some of the past sequence of cause-and-effect that led up to it (what 'was') and the alternate possibilities for that event that could have happened but didn't (what 'could be'). And even if Rose-as-Bad-Wolf is actually talking about all events throughout spacetime, one could interpret the Doctor's statement as meaning that he has the same type of awareness about any specific event in his vicinity, without this awareness being as broad as hers.
Consider this similar statement from the transcript of "The Fires of Pompeii", in which Donna is trying to convince the Doctor to save the population of Pompeii from being killed in the volcanic explosion that's supposed to happen the next day:
DONNA: But I'm history to you. You saved me in 2008. You saved us all.
Why is that different
DOCTOR: Some things are fixed, some things are in flux. Pompeii is
DONNA: How do you know which is which?
DOCTOR: Because that's how I see the universe. Every waking second, I
can see what is, what was, what could be, what must not. That's the
burden of a Time Lord, Donna. And I'm the only one left.
In this context it seems quite possible that he just means that when he's in Pompeii the day before the explosion, he can see the various possibilities and necessities for what's going to happen tomorrow. Or perhaps it's a bit like vision where we can choose to focus our attention on things near us or things far away, but with limits on how much fine-grained detail we can see for faraway things like distant mountains--he could have a very broad sense of the type of future that would arise if the volcano was prevented from exploding by the Pyrovile aliens in that episode, without seeing the details of every life and every event in that possible future.
This is all fairly speculative since I don't think the nature of Time Lord "vision" has ever really been spelled out (maybe some of the non-TV stories dealing with the concept of the Web of Time could give more hints), but the point is that both quotes are ambiguous and don't necessarily mean he has complete conscious awareness of every detail of spacetime at every moment (in fact there are many scenes of him not knowing things or arriving in a new time and being excited to learn what's out there, so I think we can safely rule out that type of omniscience).