"Two writers are writing beside the table."
Translation:Két író ír az asztal mellett.
They are two versions of the number two. "Kettő" tends to stand alone, while "két" is frequently part of some kind of a compound structure. "Két" is almost never used alone, without an object that it counts. Except when you are practicing your synchronized marching: egy... két... egy... két...
But "kettő" can usually be used in the place of "két", for emphasis, for example. Or to make sure one is not misunderstood, as "két" can be mis-heard as "hét" (seven). Think "nine-er" in military lingo.
How many pieces do you want?
- I want two - Kettőt kérek. - Standing alone
- I want two pieces - Két darabot kérek - counting the pieces
- See you at two - Viszlát kettőkor. - Standing alone
- See you at two o'clock - Viszlát két órakor - counting the hours
In several situations, not stand-alone situations, they are quite interchangeable.
- Két darabot kérek - Kettő darabot kérek - both are good
- Két óra múlva - Kettő óra múlva - after two hours - both are good
"írnak" is incorrect because the noun is singular, író. Nouns can't be plural if there's a number or another quantifier in front of it, such as "sok" or "kevés."