Chochom
I think generally nisyonos are matters of spirituality, so for example you will have a nisayon to not learn or to talk loshon hora, to look where you shouldnt, to be a baal gayva.
An onesh is a punishment, like if you get sick or lose a significant amount of money…usually Hashem does not bring bad things on a person as a test. Usually they are an outcome (middah keneged middah) for his actions. However, in every onesh there lies a nesayon. Hashem wants that you should accept the punishment as coming from Him as an atonement for your misdeeds. If you ignore the message Hashem will keep sending punishments. Hashem wants us to do Teshuvah and the punishment only works when the persons heart wakes up and he realizes Who did it to him and thinks about the why.