Under vanilla Skyrim rules, Alchemy is the game-changer when it comes to Smithing. Fortify Smithing enchantments modify a value called "SmithingMod", which corresponds to the actual Smithing skill level (or so it seems). Improvements gained from smithing fortified by enchantments only are mild because they obey the quality increase formula which you can find on UESP.
However, Fortify Smithing potions increase a value called "SmithingPowerMod". Improvements gained from fortifying this value are percentages directly applied to the weapon's damage (or armor's value), bypassing the skill-based formula, and resulting in outrageous and potentially gamebreaking weapons and armor. Whereas the armor-derived damage resistance is naturally capped at 85%, weapon damage is not capped at all.
So, to sum it up: to break your game (for fun and profit or, au contraire, for anguish and loss - your mileage may vary), brew Fortify Smithing potions.
To not break your game, don't brew them and only use Fortify Smithing enchantments.
I sort of suspect that the two different effects are a result of oversight on gamesas's part. Or maybe not. But there you have it.