Which is correct? "the traditional and the modern" or "the traditional and modern"

Both are fine. If you search for phrases like "the old and young", you'll see this kind of elision is common. (See, for example, the Forbes Article, "When the Old and Young Collide at Work" or this Business Insider article, "PEW: The Battle Between The Rich and Poor in America is Getting Fierce".)

It should be avoided where it's at all confusing. If you're using two things that aren't opposites, people might think you mean one group of things that has both properties. But where there's no chance of confusion, it's purely a style issue. Most native speakers will tend to prefer having the second "the" because it keeps the constructs parallel.

If you are going to say "the traditional", then you had better also say "the modern" so that you are referring to two sets of things.

If you say "the traditional and modern" you would be referring to one set of things that are both traditional and modern at the same time. I don't think that's your intent.