I don't think there is a TIG power source on the market that doesn't have a square wave output(?).  A square wave provides more time at peak current output by switching from DCEN to DCEP very rapidly as compared to a sine wave output.  Supposedly this aides arc stability and stiffeness, which makes sense, though honestly I can't say as I have never AC TIG welded with a sine wave output that I can recall, and therefore can't give a personal comparison.

Dig control increases the magnitude of current rise as the arc length shortens.  Basicly you get a soft arc with a buttery puddle with the dig set low - this gives excellent control for vertical welds, especially on thin material or small pipe.  Particularly nice for cover passes.  Sticking the rod may be more a problem if the welder isn't pretty steady.

Increasing Dig makes the puddle hotter (higher amperage) and more fluid when at shorter arc lengths.  This can be good for gaining reliable penetration and/or fusion.  However the additional molten metal produced by the extra amperage rise can make the puddle more dificult to control when vertical - convexity, undercutting, and burn through may be more of a problem on delicate work.

I'm not familiar with the specific ESAB machines nor the older Miller machines but the Econotig and the Hobart Tigmate (same machine) are not squarewave AC power units. Squarewave just provides much better control (more efficient use of the frequency wave) when welding aluminum.