This is a good question, because it highlights the difference between different types of "laws" of physics. There are two types of laws, roughly speaking. Empirical laws are simply mathematical rules that we come up with to try to explain patterns that we see in nature; these are often specific only to certain types of systems, and they can be proved wrong when more information and evidence is discovered. There are also fundamental laws (or scientific theories) which describe our understanding of the way that nature and the universe work at the deepest level we currently know. For example, the law that "what goes up must come down" is an empirical law; it's a "law" that describes what we observe. Einstein's theory of general relativity (that's Einstein's theory of gravity, if you haven't heard of it), on the other hand, is a fundamental law of physics; it's a fundamental description of gravity. Often, we can trace back empirical laws to more fundamental ones; for example, the law that "what goes up must come down" can be derived from Einstein's theory of gravity.

Because empirical laws don't really represent a deep understanding of the way nature works, we usually try to base our understanding and explanations of the universe as much as we can on fundamental laws and scientific theories. This is the reason that the Bode law fails for Neptune's orbit: Bode's law was an empirical law that was based on observations of the planets we knew of, but it wasn't based on any fundamental physical laws; that means that there's really no fundamental reason that Bode's law should be correct. And indeed, the discovery of Neptune disproved the validity of Bode's law. In short, the reason that Bode's law fails for Neptune's orbit is simply that there's no reason to expect that it should be correct: it's not based on any fundamental understanding of how nature works.

That doesn't mean that empirical laws are bad or useless, though: sometimes, empirical laws can help us discover more fundamental ones. For example, Kepler's laws of gravity are empirical laws about the shapes of the orbits of the planets which Kepler came up with from looking at patterns in lots of astronomical data (maybe you've heard of one of them: Kepler's first law says that the orbits of the planets are ellipses). But a hundred years later, when Newton solved for the motion of the planets using Newtonian gravity, he discovered that Kepler's laws are consequences of Newtonian mechanics and Newtonian gravity. Then two hundred years later, Einstein showed that Newton's law of gravity is a consequence of the theory of general relativity.

So, when there's a pattern in nature that can be described by a law, the interesting question is to ask what fundamental laws and principles are causing the observed law.

Bode's law is a great example of how we do science today (and then). A couple of people (really just Titius as per the wikipedia page) happened to notice that if you write down the major axis length of each planet's orbit around the sun (the major axis length is to an ellipse what the radius is to a circle) then each length is about twice as large as the previous one. This is an example of a power law (for example,. it fits the form f = a + b*c^d). Here is a general truism we have come to appreciate in physics: the fundamental physics dictates the "functional form" of the equation that describes it. Since we noticed that the size of planetary orbits follows a power law, then we should look for an explanation that involves a power law. The best reasoning I've been able to uncover after poking around a bit is something called "orbital resonance". Effectively, if you have 3 bodies swinging around each other by Newton's laws then you can show mathematically (sorry, to explain this concretely would require pages, not a paragraph!) that the orbits are unstable (ie. change in time) until a certain spacing is achieved between them. This spacing maintains a regular pattern (ie. the size of the orbits follows 1:1.5:3:6... or 1:2:4:8...). This could explain the power law and, therefore, why Bode's law works so well for the nearer planets. It's important to remember, however, that this 'law' is simply a curious pattern we noticed. The fact that it is betrayed by additional information (the knowledge of Neptune, etc) means it is incomplete.You asked why Neptune does not follow Bode's law. This is because Bode's law is not a fundamental law which, according to how we understand physics, everything must abide by. It is simply an observation. It's important to keep in mind that if you know what the equation looks like (for example, a power law like f = a + b*c^d) then you can just fiddle with the numbers until it matches your data (we do this all the time - it's called 'fitting' and it can be quite misleading if misrepresented). It is likely that orbital resonance explains why power law spacing is observed in planetary orbitals. This explains why the inner planets would follow a pattern like that observed. Since the effect of the inner planets on the outer planets is fairly weak, it's not surprising that Neptune hasn't been 'forced' into resonance. It's possible that if we studied the orbit of Neptune over a long period of time we would notice that it is approaching the distance predicted by the law (but the time scales for this are likely very, very long). The fact that Pluto follows the law if you neglect Neptune is almost certainly coincidence.

Punchline: If there is fundamental physics that demands a power law distribution of orbital widths (i.e. orbital resonance) then Bode's law is just fiddling with the numbers to fit that law, which only applies for the planets that are bunched together. As you get further from the Sun, the coupling between planets gets weaker, and correlations (how the planets affect each other to enforce long-range order) die out. If the planets were much denser so that their gravitational forces were stronger, then we would expect Bode's law to be valid further out (maybe including Neptune/Pluto/etc).

Hundreds of years ago, Bodes law was taken very seriously. Now we understand that the "law" arises as a consequence of the way the planets formed. That is, we now understand that planets formed by a process called collisional accretion. Basically small particles collide with each other and fuse to form bigger particles. When that happens, we note that bigger particles grow faster than smaller ones due to their self gravity and hence sweep up all the particles that are nearby creating empty zones. The spacing between the planets is largely due to the sweeping up of nearby particles creating empty regions.

Neptune is 30% closer to the sun than Bode's law would predict. Pluto and Eris are even worse (Pluto's orbit crosses Neptune's!). Bode's law is generally not accepted anymore.