(a bit of Cubase “history” )…
That “box” in the Tempo Track Editor window, used to be where tempo editing took place. On the other hand, until Cubase 5.1, the Tempo Track Editor had no Info Line.
With Cubase 5.1 came the addition of optionally showing a Tempo Track, and Time Signature Track, in the Project window itself. It made obvious sense to do Tempo event editing, from the Project window, via the regular Info Line, while the displayed Tempo in the Transport Bar, just shows the current tempo.
So, logically, IMHO, they reworked the Tempo Track Editor window to behave the same way.
As regards “Fixed” tempo vs “Track” tempo (btw, in Fixed mode, that tempo box, whether in the Transport Bar or in the Tempo Track Editor window, is in fact still used to change the fixed tempo), it does come in useful sometimes (no, nothing to do with Linear vs Musical Timebase)… If your Project just uses one single tempo, then it avoids having to use the Tempo Track at all, but it is also very useful for, say, setting it to a fairly low tempo, so you can switch to it from Track mode, in order to play some MIDI that is too difficult at the tempo in the Tempo Track.
Also, (especially prior to the new Tempo Track in the Project window), it is the quickest way of trying out different tempos.