earlygame: fizz is much stronger than nid, especially at level 3-4, so u can easily outdamage her, but its hard to kill her cause if shes not dumb she wont let u reach her without your E, which means u cant dodge her spear which will hit u right into your face if u attack her. just survive as nid pre 6 and u will see its easy.
mdigame: get sheen first as nid is the key of winning. ONLY USE YOUR RANGE FORM FOR HEAL! dont use it for farming, because if u farm in your melee form, u can denie fizz´ farm by just spamming your no mana cost skills on him. with sheen you can trade 1:1 with him, but you have a heal while he hasnt. healthpots and junglers can change alot. but sinsce u got lich bane (should be at 16-18 minutes) , you can simply burst him down to ½ hp with QWE melee from. do that twice and hes done. when he engages, just fight him back with ONE combo, than run! you can heal up, he cant! always think about that!(blue ofc needed)
lategame: nearly equal. both are go in go out champs. u kill one, than you have to run! i dont think that fizz can reach nids dmg, but fizz has aoe ult with cc, and much better escape for a short run, while nid can escape over long ranges which isnt needed in teamfights. i think they are even champs in lategame.
dont get me wrong, i dont say that nid always wins vs fizz, jungler depends a lot here, but if we dont talk about junglers and same palyer skill, nid should have a good advantage past level 6+sheen.
hope i was able to help