These easily confused adjectives have fairly different meanings.
Noisome describes offensive sensations of any kind, and offensive individuals and events. However, it is usually used only for things that can be experienced through the senses—not for offensive beliefs:
He is a noisome individual who rarely showers and yells at people constantly.
Noisy, on the other hand, describes noise, which has two meanings—(1) a lot of loud sounds, such as at a party, or on a city street, and (2) audio or video interference, or “static.”
I cannot sleep in noisy buildings where people play music late at night.
Early radio and tv transmissions were often too noisy to understand.
Noisy can also describe a more abstract form of static:
On the internet, it is difficult for people to filter out the truth through all of the noise of different opinions and propaganda.