In Dutch, we have just one word for pots and pans, which happens to be "pan", so I was surprised to learn about the word "pot". I learned that pans are for frying, which is why they are shallow and pots are for boiling, which is why they are tall and have straight sides. But that doesn't seem entirely correct. Saucepans seem to have the properties of a pot, but are a pan.
There is a lot of conflicting information on the internet. So what's a pan and what's a pot?
I think the answer to your question lies in the etymology of the words.
Pan is actually coming from Germaic pfanne (in Dutch panne). Which is from Latin patina (shallow pan, dish) and Greek patane (dish, plate).
On the other hand, pot also has Germanic roots it means vessel (also in Dutch) coming from pottus (drinking cup) for Latin.
So the difference as the etymology of the words put it, pans are shallow (akin to dish, plate) most suitable for shallow frying, and pot is a vessel for boiling, simmering or deep frying.
Note: Like all living things, languages can also evolve and words don't necessarily stay true to their origins. Saucepan should have been called saucepot, but that’s just how languages work; as people coining terms and words don't necessarily put much effort in the history and story of the words.
As a native English speaker, certainly where I come from, saucepan and cooking pot are mostly interchangeable. However, I understand the differentiation to be that "pans" have long, extended handles and "pots" do not. So a pan:
And a pot:
Edit to include dictionary definitions kindly provided in a comment by user3169
Pan: "A pan is a round metal container with a long handle, which is used for cooking things in, usually on top of a cooker or stove."
Pot: "A pot is a deep round container used for cooking stews, soups, and other food."
While the definition of pot specifically mentions depth, the definition of pan does not, which leads me to believe that a pan could have reasonable depth. The main distinction still seems to be the long handle for a pan.
