I think you understand things correctly.
How is used to ask about the quality/degree of something, or the manner in which something was done. When you ask someone how is X, they may respond with a details in a similar manner as what, but the reason is to communicate a quality/degree - typically "good" or "bad."
So...
How is your teacher?
She's this really weird tall lady with these crazy glasses, and she was yelling at everyone today.
The implication here is that she considers the teacher a bit intimidating, so somewhat "bad." And that's what the person asking how really wanted to know.
What's X like to me, implies that you are less interested in the person's personal feelings - rather, you want to know how X compares to other X's - than how's does. But this can be influenced either way greatly by context.
Q: What's the weather like over there?
A: It snows a lot down here, and sometimes it rains.
Q: So usually, how's the weather over there?
A: It snows a lot down here. But it's raining now.
Q: How's the weather?
A: Rainy. Take a jacket.
Q: What's the weather like?
A: Today? It's rainy.
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
To describe what is actually happening right now, you use the verb form:
To describe the sort of day it is, you use the adjective form:
In your first sentence, either rainy or raining could fit, depending on what you actually want to say; "... because it is raining" indicates that water is physically falling from the sky right now, while "because it is rainy" indicates that it is the sort of day where rain is extremely likely to happen, but doesn't necessarily mean that rain is falling as you speak.
In the second sentence, only rainy fits because it is describing a quality of the weather, not an action.