Aioli is a garlic and olive oil emulsion - or at least there is an emulsion between the olive oil and the garlic's juices. That means you start with the garlic, which will be emulsified throughout the entire Aioli. Most people add egg yolks, lemon juice, potato, or bread, to help the emulsion, and for taste.
Garlic mayo is mayonnaise, an emulsion of oil and egg yolks, with garlic added at the end. You'll have little pieces of garlic floating in the mayo but the garlic is not emulsified, meaning its taste is not present with every little drop of mayo. If that helps, I think of the difference between garlic mayo and Aioli as the difference between vegetable broth with small pieces of chicken floating around, and chicken broth.
For a good Aioli, use a mortar and pestle to crush the garlic into a paste and slowly drizzle the olive oil while you continue crushing to emulsify the oil with the garlic puree.
Here's one that was made with olive oil, garlic, and salt, nothing else:
If you want to make a garlic mayo, as opposed to an aioli, roast peeled garlic cloves submerged in oil (of choice for making the mayo) in a covered pan at 300 degrees for about 75 minutes (depending upon amount of garlic used). Strain the cloves, being sure to reserve the oil for making the mayo, and puree the cloves. Make your mayo using the reserved oil and fold in your garlic puree.
I think either confit/fresh garlic is nice. Depends what you will use it for. But anyway
My recipe would be:
Red wine vinegar
And water if it's is too thick.
Another flavored mayonnaise I really recommend is: to use roasted chili and capsicum. Just put them in the oven, and let them get nice black colour not al black but just a little here and their. Let them cool, them peel of the skin, and take the seed house away to. Then just purée it and add it your already made up mayo!