they are both nouns, but are used differently. She has a contagious laugh. Her laughter was contagious. Laughter is the sound that is made when someone laughs. She laughed at my joke. Her laughter could be heard from the other room. I don't know how helpful that is, but it is the best I can do by way of explanation.

You would say 'She has a contagious laugh'. You would not use 'laughter' here. 'Laugh' is usually a verb, but in this instance you are using it as a noun. A person's 'laugh' is their individual way of laughing. It is a personal characteristic, like their voice. 'Laughter' is a more general and more abstract noun.

Laugh is a verb. I laugh a lot! Laugher is a noun. Mykyta is laughing in a strange way, hm I can't understand his strange laughter.