The artifact feet are must for kicking WS. Relic set for asuran fists. The rest of artifact are basically macro pieces.
What Buukki said. Also, for the macro pieces the +2/+3 versions are a pretty minor improvement (if any) over the base 119 gear - so it's not really vital to have them fully upgraded for macro purposes.
Which pieces of +2/3 AF/Relic are worth getting for MNK? I've got deed vouchers to burn and am trying to split them between RUN and MNK to save myself some card farming.
Since you're comparing MNK to RUN to know which of each specific piece might be a better use of things like Deeds tokens... My take on the piece-by-piece comparison:
AF
Head: RUN
Runeist head is pretty nice for Fast Cast (14%) if you don't have Carmine +1 D head (same FC+14%), and probably the more useful piece. MNK is Focus macro (slight duration increase each upgrade from the 109 version). Either is fine, neither is amazing.
Body: RUN
RUN AF body is the star of the set, AF+3 is a default tanking body choice. Definitely go for that over the niche use of MNK AF (mostly a Chakra macro piece).
Hands: RUN
Runeist hands have Gambit duration increase, which can be important for reducing a mob's defense against a particular element. A couple more seconds of stronger damage window can be pretty relevant here when doing something like a SMN zerg. Go for that over MNK's deceptive looking WSD+10% hands (like Bippin says, mostly a Final Heaven piece only).
Legs: MNK I guess, but neither are impressive
Neither RUN nor MNK are that great. A lot of people inexplicably tend to like RUN legs for resist sets even though they're worse than Turms for Meva and worse than Erilaz +1 for combo Meva/DT-. MNK is super niche though, main use is if you really want to mess with MNK counter sets (like, maybe you have a Spharai).
Feet: MNK
Clear choice here, MNK feet are phenomenal for all things kick (including WS). RUN feet do have some use for Pflug potency increase, I just think MNK are better.
Relic
Set-wide note that especially if you're using Karambit, MNK relic gets some bonus points across the board since 5/5 pieces are great for Asuran Fists.
Head: RUN
Easy call, RUN head is incredible. A staple tanking headpiece with PDT-6%, Phalanx +7 is wonderful, and enhances Battuta is vital for a key JA (though the enhancement is the same from lv90 on up). MNK head is a Chi Blast/Penance macro piece, and a niche WD+10% (Final Heaven) piece.
Body: RUN
Another no-brainer, RUN body has DT-9%/Regen+5 and is a great idle or tanking option, and you need to at least have the lv90+ piece for Elemental Sforzo and Liement macro. MNK body is super-niche, basically Formless Strikes duration macro gear and lv90+ is fine for that purpose (if you even have FS merited any more).
Hands: MNK
Neither are amazing, but Chakra macro gear (stronger healing, remove more effects) is nice - I pretty much just use AF body/Relic hands for Chakras that refill all HP. RUN hands are rather weak, they don't really have any niche.
Legs: MNK? (but this is a close one and YMMV)
Unlike AF legs being a toss-up because neither are terribly useful, this one is a close call because both have solid uses. RUN has enhancing magic duration+ and is a great precast enhancing magic piece. MNK is great for kick sets and it's a must to get at least the lv90 version for Hundred Fists duration.
Feet: MNK
Hesychast feet have fairly limited niche use: counter sets, Asuran Fists, and Mantra macro (lv90+). But Futhark feet are really just a Rayke macro piece and don't require +2/+3 version to do that.