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As far as I know both words could be used to describe usefulness/handiness/convenience of something.
What are the main differences between them? In particular, what are the use patterns for these words? What would be the example sentences that demonstrate the word usage.
Are they form a formal/colloquial pair or they behave like 大事 and 大切?
The difference is pretty clear cut and I think they map pretty closely to the English usage of "useful" and "convenient," respectively. This is one that the dictionary will illuminate for us, so if I may...
有用:
役に立つ・こと（さま）。
⇔無用
「国家―の人材」「―な品物」「社会に―な人材」
If you look purely at the kanji you can get the idea: something that has use. The definition for it is just an alternate way of saying "useful," or 役に立つ. The examples from the dictionary can give you some idea, but searching alc will show you with a lot more examples that this has the general meaning of being of use.
Now for the other contender:
便利:
都合のよいこと。役に立って具合のよいこと。また、そのさま。
⇔不便
「通勤に―な土地」「―な道具」「この辺は何かと―がよい」
It means "convenient," plain and simple. It is similar to 有用, yes, and indeed both definitions contain the phrase 役に立つ, but 便利 expands on that a little bit to be something that improves conditions in some way rather than just being a thing that can be used. To clarify a little more on the difference, something that is 有用 is something that can be utilized to achieve some good or some goal, but something that is 便利 is inherently beneficial in some way in making some situation better.
For example, you could say 有用な人, or someone we would refer to as a "valuable asset" to a team or whatever, but we would not call this person convenient in the same context.
