What is the difference between a thickener and an emulsifier?

As Max points out, emulsifiers work by allowing two normally incompatible ingredients to mix. There are different ways that emulsifiers do this. Lecithin, probably the most common emulsifier, can do this because its molecule has a water-binding end and an oil-binding end. Hydrocolloids, like xanthan gum, can also have emulsification properties, but they work quite a bit differently.

Hydrocolloids work by binding up water, which causes a marked increase in viscosity. This allows them to function as thickeners. They are sometimes used in this capacity in combination with other emulsifiers as the increased viscosity helps to make emulsions more stable. Some hydrocolloids can work as emulsifiers on their own because they either increase viscosity enough that the droplets of differing materials can't separate, or they have a small protein component that helps to bind the ingredients.

In your case, the main difference if you switched ingredients would probably be texture. Different hydrocolloids can have similar thickening capacities, but give markedly different textures. Because of this, they are frequently used in "synergistic systems", where multiple hydrocolloids are used to modify each others' textures to acheive a desired result. This is basically what's happening in the commercial sauces.

If the xanthan gum worked fine for your purposes on its own, I'd just stick with that. Modified starch can be a lot of different things with different purposes, so if your problem is solved more simply, it's probably best not to go down that rabbit hole.

Emulsifier will make two or more element blend together, for example the egg in the mayonnaise recipe will act as an emulsifier.

A thickener will simply make something thick, as you wrote, cornstarch is a thickener.

Personally, I would use neither cornstarch or xanthan gum in a chill sauce; seems to me that it is a shortcut instead of letting it cook down properly (but I might be wrong).