In the U.S., at least, either "laundry" or "wash" would convey your meaning. I think "laundry" is more commonly used. Either can mean the clothes themselves, or a place or state where clothes go for the cleaning process.
"Please put your dirty clothes in the laundry."
"Please put your dirty clothes in the wash."
"I put the laundry in the washing machine."
"Sally folded the laundry."
"Put the laundry away."
But once you put it in the drawer or hang it in the closet, it's not laundry any more; it's just clothes!
There are at least two ways these terms can be used:
1) They can be used to indicate assent:
For example:
We need some more eggs to make the soufflé. Can you go buy some?
Alright. (or, Okay.)
2) They can be used to indicate that the quality of something is "satisfactory but not exceptionally or especially good":
For example:
How was the movie last night?
It was okay. (or, It was alright)
In these two contexts, I find these terms pretty much interchangeable. In the first, you agree, but without any particular eagerness. In the second, they both convey about the same sense of enthusiasm (or apathy).
I will offer an opinion that differs slightly from Mowzer's: I think they are both somewhat general and informal terms. If for some reason more formality, politeness, or precision is called for, I would recommend using different words altogether. For example:
Judge: I'll need you to approach the bench and take the oath.
My response: Yes, your honor. (not, "Alright" or, "Okay")
English Professor: What did you think of the Faulkner story you read this weekend?
My response: It has a good message, but it's a little dry. (not, "It's alright" or, "It's okay")
Girlfriend's mother: We need some more eggs to make the soufflé. Can you go buy some?
My Response: I'd be glad to. (not, "Alright" or, "Okay")
I think your question is a fair one, and your analogy is a good one. But, in this case, I think the two words are much more interchangeable than, say, broke and bankrupt.
Best Answer
can be said to children when they say naughty words and was a form of punishment to "wash" the filth out.
An adult version is
Your second phrase
can all be used in the same circumstances as "wash you mouth" and is a strong warning not to continue speaking in either the same way or in the same vein about something.