TRON is a multilingual character set designed in Japan as part of the larger BTRON system. This will largely be a rant, as I can not speak fully about the system; why should become obvious shortly.
Why TRON will never surplant Unicode
Let's start with the social reasons:
Social reason 1: It's all in Japanese
I have nothing against Japanese as a language, but it's not a language everyone is fluent in. In fact, besides the Japanese themselves, not very many people know Japanese. Compare this to English, French or Spanish, which are learned by a lot of students that don't have that language as their native tongue. Even Latin, Esperanto or Latino sine Flexione would have been a token effort towards Indo-European language speakers.
Social reason 2: Stop whining.
A substantial part of every English TRON webpage seems to be given over to telling us how Unicode is made by American Computer Companies and how they are trying to oppress the poor Japanese. (Occasionally, we get to hear about how the creators of Unicode are Sinocentric, too.) All of which is fine and good, but honestly, most of us don't care. Most of us are interested in how well it works for me and my projects; telling us how well it supports Hindi and Cherokee and mathematics and APL is more likely to catch the world's attention then whining about Japanese repeatedly.
Social reason 3: No one likes Japanese computer companies any more than American computer companies.
The argument that Unicode is the product of American computer companies has some merit. But Unicode can point to its close association with ISO; and while the list of full members is packed with American companies, the list of associate members includes groups like "Hong Kong Telecom CSL" and "Government of Tamil Nadu, India". Where are the non-Japanese groups involved in TRON? As far as I can tell, there are none. Judging from their webpages, I get no impression that they have any interest in supporting anything besides Japanese.
The technical reasons
While I'm not as qualified to speak on the technical reasons, not reading Japanese, I will give it a try.
Techncal reason 1: TRON does excessive blanket copying
This one has many twists and turns. Starting off with a minor social reason — saying you support me (everything besides Chinese, Japanese, or Korean) by including most of Unicode does nothing to induce me to switch from Unicode. Especially when you mention Unicode 2.0, which is missing many fun things present in Unicode 3.2.
Also, copying stuff does not make for good integration. TRON does not support combining characters (more on that below; what happens to the combining characters in Unicode? What about the languages that weren't explicitly supported in Unicode because they could use combining characters, like Sioux or Lakota or Navajo or the student orthography of Lithuanian (with additional accent marks)? Is it updated with Unicode (apparently not)?
Unicode has this problem to some extent, as some Unicode blocks are copies of old standards (ASCII became Basic Latin; ISO-8859-1 became the Latin-1 block; ISO-8859-7 became the basis for the Greek block.) TIS-620 became the basis for the Thai block, meaning that Unicode adopts the typewriter order for Thai characters, instead logical order, which is used for the rest of Unicode. But they were copied into Unicode and became part of Unicode, subject to all the rules thereof, and nothing too awful was permitted. For example, Unicode followed ISCII, but made a seperate copy for each script instead of following ISCII's lead in using script selecting control characters. Unicode also never tried swallowing 8,000 characters without change, like TRON did by accepting Unicode.
TRON repeatedly attacks Unicode for Han unification, claiming that it was a decision soley made for making it simpler for the computers. (While most Unicode gurus will argue long and hard for Han unification, they will admit that starting out with 16-bits was a simplification, that they knew would have to be worked around (and has been). They would however argue that it never could have succeeded if they had said 32-bit from the start.) This, however, seems to be a similar decision on TRON's part. Combining characters are necessary for many languages in the world. While precomposed characters (at least a thousand) could be added for them, this is not true elsewhere. Both mathematics and linguistics use the ability to add arbitrary combining characters to any character, and can't be reasonably supported by adding precomposed characters. The sole justification for this appears to be that composing characters violate the priniciple of fixed length characters.
Technical reason 3: Redundant characters
TRON has apparently encoded the same character repeatedly by reference to other character sets. This seriously hurts searching and fontmaking.
Technical reason 4: Poor non-Japanese support
The only non-Japanese characters mentioned on the English webpages are Braille characters. Thomas Chan of Cornell believes that TRON is missing even the most basic Cantonese and Hong Kong Chinese characters (of which there are several, because they aren't included in the very basic Chinese character sets that TRON references. Outside Braille and the CJK character sets that TRON references, there's no evidence they went any farther then referencing the Unicode book on this subject.
Conclusion
I could go further with the technical reasons, but I lack sufficent solid evidence. Honestly, I think the social reasons are more important; if developers seriously interested in your product can't get past them, how are the technical reasons going to come into play?
For all the claims about being a multilingual character set that will take over the world, I doubt even the developers believe it, except possibly a few with a Japan-centric view of the world. Compare TRON's rants about Americans to Unicode's "When the world wants to talk, it speaks Unicode" (an old slogan, but memorable). You can catch more flies with honey than with vinegar. Unicode also works with ISO 10646 and various governments (Iran!) and groups to produce something that is obviously more than merely the work of American computer companies with no respect for the languages. TRON is preaching to the converted, and that's not the way to overtake Unicode.