Although it seems similar the cold sores don't contain actual pus but the cold sores do contain a clear fluid in them that can seem like pus.
Pimples do have pus in them though but cold sores do not.
Cold sores are caused by the herpes simplex virus while pimples are caused by bacteria so they're different and contain different fluids in them.
If you have a cold sore you can make the cold sore go away faster by applying Apple Cider Vinegar to the cold sore 2 to 3 times per day until it goes away or you can also apply some toothpaste to the cold sore.
The toothpaste will dry out the skin and cause the cold sore to go away.
The Apple Cider Vinegar will kill off the viruses that are causing the cold sore since it's Antiseptic and Antibacterial.