Within or inside?

Both can be used as prepositions or adverbs, but only inside can function as a noun or an adjective.

So for example:

• You can talk about the inside of something, but never its *within.

• You can have an inside job, but never a *within job.

They're essentially the same, with subtle differences in connotation and approach only. Both words mean that a thing rests/resides/is located on the interior of another thing, typically a container or vessel. However, the usage differs by meaning. "Inside" tends to convey a degree of isolation or separate connotation from that which surrounds or contains it, as a spacial indicator, meaning that it and its vessel are two different, separate entities(such as a person and a building). "Within" carries the alternative connotation, that the thing contained and its vessel are closely related, or tied somehow(such as one's soul and their body). The differences are delicate, and typically ignored, aside from preference and formality.

To say 'they are the same' would mean they are interchangeable and in 90% of the cases they are not. The only time within can replace inside is in highly formal context - ' office assistant is needed, apply within'. But most frequently within is used for its own sake to mean 'in limits/reach of something', e.g. - within a hundred metres of each other, within walking distance, within someone's reach, etc..

Both words refer to the interior of a three-dimensional concave surface; i.e, a container.
But they don't refer in the same ways; there's a lot of differences. @tchrist pointed some out.
Others include:

• because inside is a noun (but within isn't), one can construct complex prepositions like
  glue the paper to the inside of the box

• similarly, because it is a noun, inside can function as an adjective, as in the inside front cover, or in noun compounds like inside man or inside information.

• inside, but not within, often uses of to mark its object:
  He got here inside (of) a minute but not *He got here within of a minute.

• inside contrasts with outside, but within contrasts with without.
  These are quite dissimilar contrasts. For one thing, they don't do negation the same way.

  • They stood there within easy earshot, but without any comprehension at all.
  • Outside of a dog, a book is a man's best friend. Inside of a dog it's too dark to read.
    –   Groucho Marx

In addition to the useful answers here, there is one very specific difference between inside and within. You are unlikely to find this difference in a dictionary.

Both inside and within are loosely said to introduce locative complements. However, the term locative is quite a loose term. We can distinguish between locations, goals and sources. In modern English both inside and within freely indicate locations. However, inside can also frequently indicate goals, whilst within rarely can. For this reason the following sentences are likely to be given very different interpretations:

• Bob walked inside the building.
• Bob walked within the building.

Notice thought that this is not an absolute difference, but a tendency. If the verb and context make it clear we can occasionally use within to indicate a goal, although this is quite rare:

• He went within the building.

To my modern English ear the sentence above is quite awkward.