TL is great. I used it for years. VT (Verituner) takes TL to the next level: TL produces an idealized, smoothed curve. VT measures IH for EVERY note and calculates a non-idealized, non-smoothed, "jagged" curve, which, contrary to what one might think intuitively, is a much better approximation of the best targets for each note of your piano. VT WILL sound better, as in "more natural," than TL. Indeed, it will sound MUCH more natural than TL.
However, if you have my experience, you will find that TL--even with its idealized (ie artificially smooth) curve, will not get much changed by a professional tuner. I ran that experiment 3 times with TL. I did a TL tuning of my Kawai 500 grand, then had a trained and vetted tuner go over it afterwards. I watched the tuner like a hawk. He didn't make many changes. That's because TL already provides the stretch; and often that is closer (sorry to say) than many tuners will have time to do using their ears.
VT, with careful--let's say fastidious--IH measurements will produce spectacular tunings; the kind of tunings that will make a good piano sound out of this world. The first thing you'll notice is RESONANCE, beyong anything you may have heard before. The second thing you'll notice is that many difficult intervals sound much, much better, or more "musical". Rolling thirds, which in equal temperament are not easy to take, will be predictable in their dissonance.
In this instance, you really do get what you pay for! VT is pricey. But I (personally) like what I get.