Classically an aioli is a garlic mayonnaise, nothing more nothing less.
It has since been bastardized into many interpretations (pesto aioli, sun-dried tomato aioli).
A remoulade is a mayonnaise with mustard, capers, chopped gherkins, herbs and anchovies.
Volupta is correct, Aïoli is classically just garlic and olive oil. You can add yolks, lemon juice and/or potato to help the emulsion, but it's not necessary (and according to purists it's no longer an Aïoli if you do).
Remoulade is a mayonnaise with mustard and garlic - and possibly herbs, hard boiled egg yolk, capers, cornichons, anchovies...etc.
“ To one pint of Mayonnaise add one large tablespoon of prepared mustard, another of gherkins, and yet another of chopped and pressed out capers, one tablespoon of fine herbs, parsley, chevril, and terragon, all chopped and mixed, and a teaspoon of anchovy essence or a bit of anchovy paste. “
“ Chop one oz. Garlic cloves as finely as possible, and add the yolk of one raw egg. A pinch of salt, and one half pint of oil, letting it trickle in a thread and stirring meanwhile, so as to effect a complete cohesion of the mixture. Add a few drops of lemon juice and cold water to the sauce as it thickens, this being done to avoid its turning.”
Well, a major difference is that they're spelled differently.
Sorry, I couldn't resist. /img/vbsmilies/smilies/cool.gif
Now, for real.
While both are classic mayonnaise sauces they are prepared differently.
Aioli is made by combining garlic and egg yolks (traditionally with a mortar and pestle) and working in olive oil a drop at a time. Water and lemon juice can be added anytime after the emulsion is halfway mixed.
Key point: the garlic is an integral part of the emulsion.
Sauce Remoulade is prepared by first making a basic mayonnaise, and then flavoring it with the other ingredients---which include mustard, capers, anchovies, and fresh herbs. A later modification uses celeraic as the flavoring agent.
Key point: The emulsion is prepared and then the flavorings added to it.