Weapon upgrades, cool.
Does this mean that the buster sword doesn't become really weak a few hours into the game?
Thats right, each weapon has its own characteristics and grows through upgrades. So no, the weapons never become obsolete.
Each weapon will lean in a certain direction (i.e. balanced, defensive, offensive, battle mage, healer, speed, etc). You can also tweak the weapon by choosing certain abilities and each weapon comes with a unique weapon skill that the character can learn. It allows you to load out the characters to your own playstyle or strategy. The characters are still built in a certain way (Cloud middle of the road, Tifa speed, Aerith magic, Barret ranged/tank) but you can tweak them in a certain direction.
So, the weapons are for that rather than replacing the previous. The customisation system is cool.