Mentor and Geist are pulling for the same slots, and I think using either optimally means molding your deck toward their strengths.

Geist is better in the matchups with less creatures since his downside of being blockable is mitigated by your ability to remove as many creatures as they can block (or kill in response to blocks being declared). An optimal geist-centric deck would be built toward maintaining board control with your resources and allowing it to swing without dying as frequently as possible. I think this style of deck benefits from "hard" counterspells, as much as Modern will let you have at least, like Deprive and possibly Spell Snare if you expect a lot of Snapcaster shenanigans to block Geist. Allow the threats you can kill to land on the table, then swing with geist a few times and counter anything scary.

Mentor is better in creature-centric matchups and always has the ability to take over any game quickly, which is a desirable trait in my eyes since I always think viewing a deck in the, "What's the most broken thing this deck can do," light is important in modern. Insane draws regularly happen (1-2-Karn!) and being able to have your own ability to top deck out of a bad situation is almost necessary. Geist doesn't really give you the immediate impact that Mentor can, if you're holding back spells. That's not particularly hard to do since your hand hate is useless late game and this converts it to pump spells and tokens.

If you're going Mentor I like moving to more cantrips and even maindecking a couple Dispel isn't the worst thing in the world (which doubles as blowing out Cryptics in the control matchup -- one you lose some headway on by dropping Geist). Extra cantrips and 1-cmc spells are great for Tasigur, too, so I think there's some synergy to be had there. I don't think I'd play Sorin or Geists if I went with Mentor since you'd want to load up on instants and sorceries.

Liliana otV I don't really like as much as Liliana tLH, who helps with Tasigur, Snapcaster, and bringing back threats. LotV also, as you said, sucks with counterspells and would force you to reshape your deck a lot.