Some

"Some", in its weak, unstressed, and most common form, basically means "a certain quantity/number of" (something). In this case, the vowel in it is hardly pronounced (sᵊm). Example: "Do you want some bread?" (« Est-ce que tu veux du pain? »). It is a very neutral particle, a vague quantifier. It means neither a lot, nor a little. It just posits the existence of the thing that is named. Note, however, that a positive value is conveyed by "some" in this case. If you ask a friend or relative "do you want some bread", it means that there actually is bread on the table, and that the person may have it if s/he wants it.

LAST BUT NOT LEAST, "some", in its stressed form (sɅm), has a qualitative value and means "a remarkable kind of". Compare, for example, "Do you want some [sᵊm] tea ?" (neutral) with "Oh, that expensive Da Hong Pao was some [sɅm] tea, indeed!" (it was a hell of a tea, the greatest tea I’ve ever tasted, etc.). Normally, context shows you the value of "some".

Any

On the contrary, "any" has a slightly more negative meaning, and is used for virtual/yet unrealized things. Thus, you would ask somebody "Do you want any bread?" if there is no bread on the table, and you would need to go and grab some in the kitchen especially for him/her. "Any" derives from "an" which itself derives from "one", and typically means "at least one". In a certain category of items, "any" selects at least one instance of those. So the meaning is slightly negative, less affirmative.
Examples:

  • a) "Is there any way that I can help you?" (hidden message: I’m not sure I can really help you). 
  • b) "Is there some way that I can help you?" (hidden message: there must be a way, and I really want to help).

In French, a) could be translated as « Y a-t-il quelque chose que je puisse faire ? » and b) as « Qu’est-ce que je peux faire pour aider ? ».

"Any" also means "any one instance/example of" and is used broadly to say "any kind of".
Example: "Any French person has eaten baguette at least once in their life." (i.e. whoever you stop in the street in France will have eaten baguette at least once).

Another related case where "any" is used is when the speaker refuses to make a choice.
Example: "Which pen color would you like? Give me any color, I don’t mind" (in French, « n’importe lequel/laquelle »)