Here is the definition of the word within from Oxford Dictionary:
So does that mean the two words have no difference, and can be used exchangingly? Is there any connotation that one word can be used but not the other?
Inside may be used while describing position/location as opposed to outside. Example: The box is blue colored inside and black outside.
Within may be used for time/space. Example: I'll reach there within ten minutes. Stay within the boundaries.
Like any two synonyms, there will be contexts where you can use either one, and there will be other contexts where most people would choose one word over the other. These two words are such general-purpose words (with some idiomatic usages as well) that I don't think we can begin to cover all the bases (that would not be within reason).
That said, here is one example where either word works just fine:
There were many troubles hidden inside Pandora's box.
There were many troubles hidden within Pandora's box.
Here's one place where I'd clearly use inside:
After the rain started, we went inside the house.
and another where I'd clearly use within:
Would Freddy be traded to another team? That was within the realm of possibility.
You can always use the Ngrams tool to check if a certain usage can go both ways, or if it generally doesn't. The Ngrams tool is not infallible, but, if you have an internet connection, it's always within reach.
There are a lot of external links embedded within this answer; I hope they lead to some fruitful research for the O.P.
Their meanings are similar, but they are likely to be used in different contexts. Within is perhaps a little more formal and dated. There is an important grammatical difference you need to be aware of. Within can be used only as a preposition and an adverb. Inside can additionally be used as a noun and an adjective.
They more or less mean the same thing.
Inside is more literal, one thing is in the other thing, like in a box.
Within doesn't necessarily mean "inside". You buy a VCR, and there's the user manual included within. That doesn't mean you need to unscrew the cover or take it out of the battery container of the VCR. It means it's in the box, included with the VCR. You buy a computer magazine and there's a CD included within - but the CD is in an envelope glued to the front cover, not inside the magazine. Of course "within" may as well be "inside" - you buy a toy, and "3 AA batteries included within" - they may be in the box, and you need to load them into the toy, or they may be already loaded and inside the toy. Anyway, "Within" is the middle ground between "Inside" and "Bundled" (the latter meaning included in the set, but definitely outside of the core item)