Charles, both spellings and several others are from the same roots (see http://fitzsimons.info ) and an article from Mick Fitzsimons (small "s" amd one "m") who has an interesting article here
http://fitzsimons.info/one-m-or-two.html which contends that the double-m spelling is predominantly from the English side.

Only today my son Aaron FitzSimons found his name misspelled in today's paper by a journalanist who despite Aaron spelling it as above and stressing the upper case "S" one the one "m", she still couldn't help herself and spelled it by the most prevalent spelling as "Fitzsimmons".

I have had a lifetime of this indeed not only in spelling it, but by pronouncing FitzSimons (like Fitz-Sigh-mons) only to be immediately corrected by them "OK Mr Fitzsimmons". It's a lost battle Charles. :-)