Newton's describes his notion of absolute time and space his Scholium on Time, Space, Place and Motion.
In Newton's time, civil time was still measured by the motion of the Sun. Newton needed to distinguish time as measured by a sundial from the time as measured by a clock (or by the motions of the planets, or of Jupiter's moons). Scientists in Newton's day were well aware that time kept by a sundial and time kept by a pendulum clock didn't agree. For example, the 24 solar hours between solar noon on December 26 and solar noon on December 27 are 30 seconds in excess of 24 hours as measured by a clock. Between October 20 and October 21, it's 22 seconds shy of 24 hours. Newton viewed pendulum clocks as yielding a truer measure of time interval than time as measured by a sundial.
In addition to arguing against sundials, Newton was also arguing against Descartes in this scholium. Descartes' Principia Philosophae (1644) put forth a rather different world view than Newton's Principia Mathematica. A number of Newton's arguments were veiled attacks on Descartes' concepts.
Newton implied a lot more than this in his scholium. He apparently truly did believe that there was some unknowable (except to God) absolute time and space, God's frame of reference. Newton saw signs of this in (for example) his bucket argument.
The modern view is that Newton's concept of an unknowable absolute time and space is metaphysics, not physics, and is not needed in classical mechanics. Parsimony suggests that those extraneous elements be omitted from teaching and use of Newtonian mechanics, and indeed, modern classical teaching typically does not introduce these concepts except for historical reasons.
What are the flaws in the definition that make it fail?
None, in the context of classical mechanics. It is however chock full of Newton's religious views, it is unknowable, and it is not needed. The latter two (unknowability and unnecessity) are strong arguments for getting rid of it; it is not parsimonious. Lack of parsimony however does not equate with failure.
What about Newton's bucket argument? That argument is indeed a valid argument for acceleration being absolute in Newtonian mechanics (which it is; acceleration is the same in all inertial frames), but it does not show that time and space are absolute.
What makes absolute time and space fail is general relativity. Even the related concept of relative time and space as used in classical mechanics fails in this regard.