I've always found it a bit odd that Tamiya even sells a "lacquer" thinner because the only product in their range that needs a lacquer thinner is their Liquid Surface Primer (and perhaps, their putty).
That aside, it works well with Tamiya acrylic paint.
So what difference does it make?
1: It extends the drying time of the paint, giving it more time to level after application. This is particularly helpful with gloss colours, which can have a tendency towards graininess when using other thinners (including their own X-20A) The paint lays down wetter and dries to a glossier sheen.
2: You can thin the paint out far more than you can with X-20A. Most people are reluctant to thin the paint too much so as to minimise the number of coats to achieve the desired coverage. Sure, you can do the "one coat" thing with Tamiya acrylics, but the true beauty of the paint comes from its versatility. Tamiya paints can be run extremely thinly, both for general coverage and for effects (shading, glazing etc) . For many years, I've seen people recommend to thin Tamiya acrylics about 2 parts paint to one part thinner, or sometimes 1:1. However, the secret to achieving silky smooth finishes with Tamiya acrylics is to go thin, very thin. Start at about 2:1 thinner to paint and go upwards.
The problem with this is that the proprietary X-20A thinner doesn't really like to play with the paint past about 3:1. It will mix, but past that ratio the paint tends to start to lose adhesion. It will bead and run when applied (similar behaviour to when thinned with plain old water).
This is where the lacquer thinner comes in. It allows you to super-thin the paint without losing adhesion. You can take it to 10 parts thinner to one part paint and it will still stick.
Why would you want to do this?
Applying multiple thin coats of paint is always preferable to one heavy coat. Tamiya acrylics have a fairly heavy pigment load and it doesn't take as much paint as one would think to achieve coverage.
Also, there's a technique that can loosely be called "glazing". You can run the paint extremely thinly (I'm talking about in the 80-90% thinner range) and apply very thin "tint" coats to base colours (similar in concept to overall "tint" washes) in a very controlled manner for shading/fading/weathering.
Another benefit is that it allows you to get much better definition. Run the paint thin, use lower pressure, get up close and personal and get sharper lines with less overspray.