What are absolute space & time? Why are they not absolute in reality? As said by A.P. French in Newtonian point of view:

Space is a sort of stationary three-dimensional matrix into which one can place objects or through which objects can move without producing any interaction betweeen the object in the space. [...] Time, in Newton's view, is also absolute and flows on without regard to any physical object or event. [...]  One can neither speed up time nor slow down its rate, and this flow of time exists uniformly throughout the universe.

So, what do these definitions want to tell? What are absolute space & time actually? Why are not space and time absolute in reality? What are the flaws in the definition that make it fail?
 A: That's a very common misunderstanding of classical mechanics. The theory doesn't make ANY statement about the microscopic structure of space or time, it only makes a number of symmetry statements. The mechanistic view of what the theory "means" is a philosophical construct completely external to the theory. It's metaphysics, not physics. 
Time, in the theory, is simply the stream of numbers shown by the faces of a sufficiently precise clock. The clock is simply a measurement device that is assumed to exist (otherwise one couldn't talk about its values). Similarly, space is simply the readings on rulers, which are also assumed to exist. Nowhere does the theory make any statement about what causes rulers to exist or why rulers are reliable markers for space. 
The only thing the theory says about the "absoluteness" of space and time is that ALL possible statements about the dynamics of bodies that can be made in the theory depend exclusively on the differences between two times and the differences between (at least) two ruler readings that start in the same coordinate.  
That's basically the core of Galilean invariance. Everything that matters can be expressed as differences.
Strictly speaking the theory makes a third statement, which is usually suppressed (or forgotten): when angles are of importance, all relevant physics can be expressed as the difference between angles. 
Taken together these statements account for the homogeneity and isotropy of space and the homogeneity of time, which, per Noether's theorem, equal the conservation of momentum, angular momentum and energy. 
So quite to the contrary of the naive metaphysical reading of classical mechanics, the theory itself is perfectly relative. At no point does it admit absolute definitions of space and time, otherwise the conservation laws that follow would be broken, which has never been observed. 
A: 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.
