Item Level or iLvl of gear determines that gear's strength. Gear with a higher iLvl will essentially always be better than gear with a lower iLvl. This is especially significant at level caps, including former ones, since while gear could no longer increase in level, it could still grow in stats as reflected by its higher Item Level.
Player characters have an average item level that is the sum of the ilvl of all the equipped pieces of gear divided by the number of slots. Unless the player is equipped with a Shield, the main hand weapon's ilvl is counted twice in place of the empty off-hand. Thus, when trying to reach a required minimum item level for a duty, it can be best to upgrade the weapon first, even for tank or healer classes.
A notable caveat is with craftable gear. Only High Quality gear has appropriate stats for its Item Level. Any (NQ) Normal Quality gear has a 10% penalty in stats versus same-ilvl Aetherial, Dungeon, or Tomestone rarity gear. This is only a critical issue starting at level 50, once the stats on gear get larger than single digits. NPC vendors only ever sell (NQ) Normal Quality gear; the sole source of High Quality gear for Gil is from other players via the Market Board. Leveling gear can also be acquired High Quality from Side Quests that award coffers.