I am not well-versed in the proper technical terms, but the comma does indeed change the meaning.
The comma separates the sentence before it, so "which" refers to the whole separated sentence ("this position [...] invaluable experience"). Withouth the comma, the "which" only applies to the immediate subject ("invaluable experience".
It can be seen more clearly if you rewrite your sentence:
Case 1, with the comma:
"This position offers a unique opportunity to use and hone the skills that I have and gain invaluable experience. I hope will be of benefit to your company and myself."
Case 2, without:
"This position offers a unique opportunity to use and hone the skills that I have and gain invaluable experience. I hope that experience will be of benefit to your company and myself."
Notice the "that"?
In your original sentence, you can replace "which" with "that", which will be referring to the experience itself more clearly and not require the comma.
You will also notice that this will no longer be your intended meaning. So you let the "which" be and place a comma.
That all being said, your sentence is fairly long and complex. Since you mention that it is for a cover letter, you might want to use shorter sentences. These letters (and resumes) can and often are glanced over instead of being thoroughly read, in which case the simpler language is more likely to get the point across.