What's the difference between come in and come inside? When it is raining, a mother would say her daughter, come inside but when a guest comes, come in, please. I think the former is a polite way while the latter is like an order. Am I right?
It's all about tone and usage and nothing to do with the words themselves.
It would be just as polite to say either of these two phrases:
Please, do come inside.
Please, do come in.
And just as much of a command to say either of these two phrases:
George, come inside, now!
George, come in, now!
Though, I admit it sounds a tiny bit odd to use "in" in the second example but I believe this is due to the phrasing. Saying something like this is perfectly appropriate:
George, get in the house right this second!
They can both be used interchangeably. "In" is just being used as short for "inside", however neither one is more formal than the other. There is no difference in semantics between the two.
Grammatically, both are imperative sentences. But practically speaking (or functionally) imperatives are not always harsh "commands." They can be invitations:
Come to Boston when you have a chance.
Come in is an invitation for your visitor to enter your house.
Come inside (right this second)!, said by a mother to her child, is not an invitation. It is an order/command.
Now, a lot has been said about "the nuance of 'in' and 'inside'," and you can read a lot of explanations there. But we can simplify things and just say Come in is idiomatic in this particular usage of inviting a guest to enter your house, and come inside is idiomatic for ordering someone such as your child, to enter your house. Native speakers would not reverse the usage of these prepositions, unless they wanted to stray from the idiomatic uses.
Another nuance is that "come inside" implies that the recipient of the request is actually outside.
"Come in" could be used when the recipient is either outside, or being invited into another space in an already enclosed area.
To a person who is standing on the porch: "Please come inside." or "Please come in".
To a person who is standing in a hallway, and being invited into an office: "Please come in."
You can use either in or inside as an adverb in the sentence presented, without any difference in meaning. However, the use of in is much more common.