The term "transwarp", as with the expression "protomatter in the Genesis Matrix", was a melodramatic bit of technobabble whipped up to suggest that the TMP-refit Enterprise was somehow getting "old", or otherwise soon to be obsolete. To this day, there seems to be no canon substance to the notion of what transwarp is, or what potential the technology could have, or how it is differentiated from "old" warp drive. No matter what any of us says or what citations we invoke, it's all speculation and interpretation.
The VOY ep "Distant Origin" seemed to suggest that the saurian Voth, who possesed this powerful transwarp ability, had passed some kind of ultra-high-tech threshold that amounted to the society establishing itself in an era far in advance of the Federation. The way Chakotay spoke about it made it sound like any society who mastered this mysterious transwarp had achieved some kind of technological Holy Grail, where space vessels travel at speeds a whole level of magnitude above what Federation-known starships commonly do. And again, it is spoken of in conspicuously vague generalities; there is little substance to explain what it means or how it differs from any other kind of FTL propulsion in the STAR TREK Universe. "VOY's "Hope and Fear" also seemed to make some suggestion that "transwarp" was super-advanced, and even displayed a new kind of visual FX, but there was no substance there, either.
"Hope and Fear", to my point-of-view, seemed to suggest that transwarp was a technology significantly different than "lesser" warp drive. While warp drive seemed to allow as starship to "keep in touch" with ordinary space (scanning, visuals, etc.), the more insular tunnels seen with transwarp seemed to suggest a wormhole effect. Then again, this didn't make sense either. Wormholes are supposedly link two distinct fixed points in space, but the transwarp conduit concept seemed to suggest that sometimes and not others. The crazy thing about the transwarp/slipstream endeavors seen in VOY suggest that Federation starship hull technology is unable to handle the stress; if this is the case for all transwarp technologies, then the much less advanced 23rd century Excelsior would never have been able to sustain transwarp velocities for any length of time.
Also keep in mind that the decree to declare Excelsior's transwarp a "failure" came from Gene Roddenberry through Richard Arnold. Whether that makes it "canon" is anybody's guess.
Here's my take on it, FWIW. This and 50¢ still can't get you a decent cup of coffee.
There is a common thread between what Chakotay was saying in "Distant Origin" and what Captain Archer and Trip repeatedly talked about in ENT: FTL propulsion in the STAR TREK Universe seems to come in stages, which becomes a multi-generational technology convention of sorts. In "First Flight", Earth scored a major victory (much to the consternation of the Vulcans) by surpassing the Warp 2 barrier. The suggestion was that Earth spacecraft had been FTL-capable for over three-quarters of a century, but that they were all limited to Warp 2 before the NX prototypes shattered that (theoretical) barrier. Once the Warp 2 barrier had been shattered, this paved the way for the final development of the Warp 5 engine. (Again, something the Vuclans took much longer to achieve, thus leading to the suggestion that Earth was moving too fast into a realm they "are not ready" for.) Apparently through interaction with other species and technologies, Earth technology continued to advance over the ten years the NX-01 Enterprise was in flight. This was no doubt helped by the boom in starship construction during the Earth-Romulan conflict and the eventual founding of the Coalition of Planets, which gave way to the newborn Federation. Archer and Tripp toasted to their own Holy Grail: "to Warp 7", "the next generation", presumably the design goal for the first generation of Federation starships. It is safe to assume that, just as Warp 5 was initially difficult for the NX-01 to sustain, so Warp 7 would be a challenge for early Federation starships.
So the tubular warp nacelle technology continued to evolve from "These Are The Yoyages..." over the next hundred or so years. During TOS, Constitution-class starships kept exceeding their design limits of Warp 8, occasionally flirting with double-digit warp factors, but at far greater risk. So, how do you achieve superior velocity without the risk of dangerous double-digit warp factors? Why, you find a way to make lesser warp factors render greater FTL speeds, of course! Here's where the non-canon fandom constructs come in handy.
If the ENT-thru-TOS-era tubular warp nacelle technology can be broadly called "circumfirential warp drive", with all of its mass-producability, quirks, and limitations, then the very different nacelles seen on TMP-era-through-TNG-era vessels can be called "linear warp drive". (Goldstein & Goldstein's 1980 "Spaceflight Chronology" suggested the Federation was investing in revolutionary technologies that would lead to super-fast, huge, super-powerful starships-of-the-future, possibly with fourth-power "Super Warp" abilities.) It's been my conjecture for a long time that the TMP-refit "linear warp drive" was the first step onto this new technological plateau. Note that in both TMP1 and TMP5, the Enterprise is never ordered above Warp 7 in canon. What's so remarkable about that if the pre-refit Enterprise repeatedly sustained Warp 8 and above? If linear Warp 7 is fourth-power, that would be (roughly) the equivalent of (circumfirential) Warp 13.39. That would make a linear-warp-driven Connie a remarkable evolution over her predecessors of the decades before. And if subsequent refits and new starship classes built upon this technology, then Warp 9 for Picard's Enterprise-D would be like Warp 18.7 for Kirk's TOS Enterprise.
The whole "transwarp" / "slipstream" phenomenon still makes no sense, unless you assume multiple technologies and techniques are at play. I assumed that the Borg, the Voth and the counterfeit Dauntless were using some kind of fifth-power technology to assume their fantastic velocities. Maybe Captain Styles and the Excelsior were trying (unsuccessfully) to achieve this feat, but could never reliably sustain it, and the technology was put back on the shelf. If TransWarp 6 were a fifth-power velocity, it would be the same as LinWarp 9.39 for Picard's Enterprise, or CircWarp 19.8 for Kirk's TOS boat.
It's far from perfect, and I have no doubt there are those who will find plenty of holes in this approach, but this kind of stair-step/plateau theory to generational propulsion technology would go a very long way to bridging the gaps.