The PHB mentions Improvised Weapons, which do not necessarily belong to either category:
Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
Natural Weapons (such as the Lizardfolk's jaws) are also a thing, and according to Jeremy Crawford they too are neither martial nor simple.
As enkryptor mentioned, some monsters use weapons that are not explicitly categorized.