be of some help / be of any help

There's no real difference in meaning.

It's possible that "to be of some help" implies optimism (that you will likely be helpful) while "to be of any help" implies the possibility that maybe you will not be helpful at all.

It's more likely that speakers will use these without any sense of this distinction though.

While there is some overlap, "any" and "some" don't mean exactly the same thing. They're both indeterminate amounts but any means any at all, even the smallest iota, while some is a more general non-specific amount. If it's something countable, then if you have exactly one, you don't have some, but you do have "any".

Usually, when some is used in a question, any is used in the negative answer: "Want some pizza?" "No thanks, I don't want any pizza." This illustrates exactly the point above.

Consider: "Have I been of any help?" (...at all? even the smallest amount? anything greater than none?) "Yes, you've been of some help" (some unspecified amount, NOT implied to be the least possible).

Practically speaking, both phrases in the original question express a similar sentiment, but do not have exactly the same meaning.