Are 汤 and 羹 used to refer to different kinds of soup or is 羹 merely a part of the full word for soup (汤羹)? For example, is one used to refer to light soups while the other is reserved for heavy broths?
In my opinion, 汤 is a more common to be seen. When it refers to "soup", the "soup" is thin. I mean, most ingredient is water, and you can find other things such as meat, vegetables,etc. inside the water.
羹 is a thick soup. Generally, we will add some 芡粉(qiàn fĕn)(most composition is the starch) to the soup(We call this action "勾芡(gōu qiàn)"). The starch will absord the water, so it makes the soup "thick".
In ancient classical texts, 汤 means hot water, while 羹 means meat with sauce(see here), and later thick meaty sauce and sticky soup.
However, when time went by, they are now all used to describe soup. But because they have different etymology, 羹 is taken as a thicker form of 汤. 汤 is a dish whose nuclear process is to boil a crock of water and consumed in the manner of drinking with a spoon. (By the way, you westerners seems to categorize things like chicken noodles also as a kind of soup....In China, soups and noodles are 2 categories excluding each other) However, 汤 is a wider concept though it does not contain noodles=,=.
In my opinion, 羹 is thicker in texture whereas 汤 is more watery like stock. for example:
chicken stock: 鸡(chicken)汤
EDIT: interesting note: when you translate the book title "chicken soup for the soul", it's translated to 心灵鸡汤 (chicken stock of the soul), so chicken soup and chicken stock both translate to 汤。