It only matters if you are min-maxing a single role and a single class, and will never, ever switch away. If you choose incorrectly, your character will be ruined for level 1 questing*. Also extremely important for role-playing purposes.
Outstanding bonuses are:
Dunesfolk Lalafell, Piety 23, highest base MP at level 1
Moon Miqote, Mind 23, highest base healing at level 1
Sea Rogaedyn, Vitality 23, highest base HP at level 1
Highlander Hyur, Strength 23, highest base melee damage at level 1
Wild Elezen, Dexterity 23, highest base ranged damage at level 1
Dusk Elezen, Intelligence 23, highest base magic damage at level 1
*Bonuses are eclipsed by gear that you obtain and can wear before you actually reach level 2 from quests that are immediately available in your starting town. Damage bonuses from a single point of strength, dexterity, or intelligence contribute about 1% of your total damage (check the character sheet if you don't believe me).
Your god or goddess also determines which elemental attacks you resist or take extra damage from*.
*Every element is encountered at least once in each zone you travel in. Every additional point or lost point in a resistance will increase or decrease damage taken by approximately 1%. Mobs regularly do less than 10 damage. Also, the game rounds to integers.
No class restriction based on race. Some races favor certain classes (if you consider an extra 0 damage at level 1 to be an advantage).
A single character can be maxed in every class. Includes combat and crafting/gathering classes. Any class that is not currently your highest level will receive a 50% bonus experience. Effectively, that means the moment you reach level 50 in one class, you don't stop receiving this experience bonus.