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.
Your two sentences have the same meaning, however the first
John thought it would be a good idea going to the museum.
is slightly awkward and would be solved by rewording
John thought going to the museum was a good idea.
John thinks going to the museum is a good idea.
use the -ing*(gerund) to describe actual events or experiences
Your second sentence
John thought it would be a good idea to go to the museum.
John thinks it is a good idea to go to the museum.
is correct and sounds natural. Use the to +verb(infinitive) to describe potential actions
Best Answer
One way to correctly understand the difference in coming and going is to understand the underlying verbs that define them. These are borrowed from the dictionary. I strongly suggest you follow and stick to one type of english because a mixture of UK-English and US-English can cause a lot of sentences to lose their meaning(personal experience).
coming(adj)-> come,came(verb) = Reach a destination by movement or Move toward something or somebody or approach something or somebody
going(adj) -> go, went,gone, goes(verb) = change location or proceed (also used metaphorically) or Move away from a place into another direction
You should notice the small differnce that come is usually used in the second line of conversation if it is a statement. If it is used in the first line of the conversation, it is usually followed by a question. example: You do not start a conversation with someone saying "I am coming to the movies." You say, I am going and someone will ask Can I come along ? or vice versa. One has to look in to the broader picture to get the meaning.
I hope this can help.
cheers pal !