Recipes can call for beef and chicken broth or stock, establishing the base for soups or sauces. So what's the difference?

Often the terms are used interchangeably, but technically, a broth is liquid made from simmering meat and/or vegetables.

on the other hand, is made by simmering bone in liquid.

When bones are simmered for a long time to make stock, gelatin is extracted along with flavor. The gelatin makes the stock thicker than a typical broth.  Roasting the bones prior to simmering gives the stock a deeper and richer flavor but is not required.

Beef stock

Beef broth

What about seasonings?

Usually a stock doesn't have much (or any) seasoning in it, while a broth is often heavily seasoned.  The idea behind stock is that it provides a clean base for adding other key flavors, while the seasoning in a broth may define the flavor of an entire dish.  This is especially the case with store-bought stocks and broths.