The new Tamron "Di III" (Digitally Integrated III) designation is for Tamron lenses optimized for mirrorless cameras with shorter registration distances (a/k/a flange focal distances) than typical SLR and DLSR cameras.
In addition to the shorter registration distances that allow for differences in lens design, the "Di III" lenses are also "all electronic" in their connection between the lens and the camera body.
With SLR/DSLR mounts, some brands also use all electronic connections while other brands still use mechanical linkages left over from designs created before the semiconductor revolution of the last third of the twentieth century.
One thing to note is that, at least thus far, most "Di III" lenses have been released for a specific mount. Some are APS-C or even Micro Four-Thirds only, others are for full frame mounts.
The Tamron 14-150mm f/3.5-5.8 Di III Lens for Micro Four Thirds is an µ4/3 only lens.
The Tamron 18-200mm F/3.5-6.3 Di III VC NEX is available in E mount for Sony NEX cameras. All NEX models have APS-C sensors. The same lens is also available in Canon EF-M mount. All Canon EF-M cameras also have APS-C sensors. This lens is currently (as of late May, 2018) the only Tamron Di III lens available in more than a single mount. It's also available with either black or silver external finish for both of the above mounts.
On the other hand, the Tamron 28-75mm f/2.8 Di III RXD Lens for Sony E is a full frame lens for the Sony E mount. Of course it could also be used on a Sony E mount camera with a smaller sensor, such as an NEX model, in much the same way that Nikon full frame Fx lenses can also be used on Nikon Dx (APS-C) cameras or Canon full frame EF lenses can also be used on Canon EF-S (APS-C) cameras. Perhaps if Canon or Nikon ever release FF mirrorless cameras with short registration distances Tamron will make this lens in those mounts as well.