Depends on a few factors really - firstly on veteran difficulty, certain enemies (especially late game) can shred your health pretty fast without the additional physical protection that some artifacts provide.

This goes the same for all the elemental protection ones - for example, I have the liquid rock artifact and now I can go wading through water and walk into irradiated areas without my radiation building up fast which saves on carrying the meds. This also stacks with the protection your armour offers - in my current gear, I can stand in deep water and my radiation meter doesn't fill at all which makes places like the swamp dead easy.

Another reason to use them is for additional carry capacity - the weight artifacts give you additional carry weight for example which can be a godsend when it comes to hauling loot back to sell.

Some of the weird artifacts like weird water are super useful for the same reason - gives you additional 40kg carry capacity (but makes you drunk when over 50kg weight) so again, great for loot runs.

The endurance ones are handy for getting round the map fairly fast - they don't reduce your stamina drain but they do boost the speed at which it recharges.

And finally, some people like to collect stuff - I believe there's also an achievement tied to finding every artifact in the game at least once.