You don't have to look too far at all to find mention of what alchemy is or to find people asking for the fastest way to level up alchemy in Skyrim, but what does leveling up the skill actually do for you? I'm not asking about the perks associated with it - What is it that actually sets a level 29 alchemist apart from a level 20 alchemist? Why bother leveling the skill up (other than for fun), if you're not going to bother with alchemist perks?
There are two things that change on what you create the higher the level: How strong the potion is, and how much it's worth. Three ingredients make stronger and more worth potion as well. If I'm not mistaken, duration and level change with level too, making a buff potion last longer, and a debuff potion\poison able to be used on higher level mobs.
When you alchemy level increases then the potions you create will become stronger. The Elder Scrolls Wiki has this table describing the levelling.