The rarity of an artifact determines the power of the effect(s) granted. Though, they follow a rather unintuitive progression. The stat-based artifacts (Colossus, Monolith, Pyramid & Spaceship) all follow the same increases, where the Rare & Epic give the same bonus - twice as powerful as a common, and the Legendary & Mythic give the same bonus - thrice as powerful as a common. The other artifacts with unique effects tend to each follow different formulae:
Bloom Filters & Black Domains can each be used on higher levels of planets; Planetary Shields give slightly exponential Defense multipliers, but the Range & Speed stats are reduced the same amounts; Photoid Cannons give a linear increase in Speed, and a linear decrease in Defense; Wormholes give an exponential effective decrease in distance.
The rarity of an artifact is dependent on the level of the artifact, which is determined by the level of the Foundry which it is found on, plus a bonus chance of being a higher level. There is a 1/16 chance that the artifact level will be +1, and a 1/64 chance that the artifact level will be +2. The artifact level is then translated into a rarity:
Common
Rare
Epic
Legendary
Mythic
Level 0 - 1
Level 2 - 3
Level 4 - 5
Level 6 - 7
Level 8+
For example, mid-level foundries (level 4/5) can still produce powerful Legendary artifacts (if the player gets lucky enough), even though the expected rarity would be Epic.