A quick google shows that your examples come from the Urban Dictionary, which dosn't always provide the best examples from which to ground a lesson, but that aside for a minute....
The term 'wash your mouth out (with soap/soapy water)' is an expression that is used humorously when someone has used a rude or offensive word. It's a throwback to a time where washing a child's mouth out with soap and water was considered a legitimate punishment for speaking rudely or using bad language. For obvious reasons, the physical punishment has become almost obsolete, but the saying still remains, and it's not uncommon to hear it used idiomatically or as a mild threat when a child has been swearing. It can also be used humourously in an adult environment, which doesn't really lend itsef well to the example you have given.
The term 'watch your mouth' also means to be careful about what you say or how you speak, but would be considered a stronger warning. While tone and context can obviously change the meaning considerably, 'watch your mouth' can be used in a far more adult, and threatening manner than 'wash your mouth'. It's not uncommon to hear the term used in London gangland movies in a threatening manner, for example:
‘you'd better watch your mouth, mate, or we'll have you’
i.e. watch what you say, or we'll beat/kill you.