Both translate as ring. Is there a difference in connotation? Or is it just a matter of dialect? To further confuse things, I have even heard anilla and aro!
Both, anillo and sortija are valid words for ring. However, sortija is commonly used for women's jewelry rings, and never for engagement rings. It has a "cheap" connotation.
Let's summarize the answer with the comments:
Anillo: The most usual way to call any jewelry ring. It can also be used with different circular objects outside the scope of jewelry.
Aro: Can be used for jewelry rings, however is not usual. It's usual scope is to name circular objects.
Anilla: Not used for jewelry rings. Usually refers to different tools of metal pieces with a circular form.
Sortija: Ring of cheap jewelry, used only for women worn rings.
Alianza: Wedding ring, for men and women, however is more usually used just for men wedding rings.
Anillo de compromiso: Engagement ring.