Using ‘on’ as the first person plural has become very common in spoken French, so common that you would be hard pressed to find anyone actually using ‘nous’ in conversation.

‘Nous’ is still however very present in written French, for example in story telling or official documents.

In summary: there is no distinction in meaning, just in formality.

I think the website is a little too strict, if they ask for “we like” they should accept both “Nous aimons” and “On aime“. If I were a teacher and I could only use one valid answer, I would put “Nous aimons“. Honestly I don’t understand why the test would be like that, but it’s probably consistent with the lessons before the test.

That being said, in spoken language, we use “on” much more than “nous“. It’s convenient because it conjugates like il/elle, which is often the shortest and easiest conjugation.

Be careful though, when nous is the subject you can always use on instead, but the opposite is not true.

The assignment was probably asking for a single sentence that was translating both "we like" and "they like”.

If this is the case, there is no other way than use on aime (or better on aime bien.)

On has essentially replaced the first person plural in spoken French but it can also replace all persons, singular and plural, depending on the context.

[Nous,] on aime s’amuser. → We like to have fun.

Alors, on aime s’amuser ? → So, you like to have fun ?

Ici, on aime s’amuser. → Here, we/they like to have fun.

On is always conjugated as third person singular because it originally means homme:

Homme aime s’amuser → Man likes to have fun.