These are quite different from each other. The Q-filter delivers more treble and tames the bass frequencies a bit, while a Varitone is an LCR-circuit which filters out mids.

The centre frequency of the Varitone is adjusted by switchable capacitors. The schematic with component values can be found on the web (e.g. here).

There seems to be different information on the inductance of the coil; some diagrams show a value of only 1.5 H. I looked at Gibson's target centre frequencies a while ago and came to the conclusion that a value around 7 H like in the Blueshawk diagram is required to achieve the desired result. Of course, one can use a smaller coil and larger caps, but the problem is that not only the centre frequencies but also the Q of the resulting LCR-circuit is important.

For these reasons, I'm a bit sceptical about some Varitone devices sold on the bay, because they're so small that they really can't contain a coil of significant inductance.

I implemented a Varitone in my semi-hollow guitar by using an old humbucking pickup as the coil (bucking hum is a nice property for the Varitone coil!). I think it sort of does what it's supposed to do, but I don't use it much. It takes away some volume and is useful for dirty sounds, but I play clean most of the time.

Unfortunately, I don't know how it compares to a Q-filter, since I never used one. HTH.
Let the thunder roll and the lightning flash, I'm doing alright for country trash.