After a bit of searching, I haven't found a clear difference between "rechts" and "nach rechts".
First of all, do they have the same meaning?
Second, are there situations in which one is clearly used instead of the other?
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They don't mean the same, they are different. It's rather simple:
Two persons watch a big old painting in an art gallery.
Oh, dieser kleine Hund ist so süß.
Oh, this little dog is so sweet.
Wo siehst du einen Hund in diesem Bild?
Where do you see a dog in this picture?
At the right bottom corner (Literally just: "right bottom")
Two persons drive through an unknown city, the passenger reads the map.
Wohin müssen wir bei der nächsten Kreuzung?
Where (to) do we have to go at the next junction?
To the right
In both of the examples given above you have no choice. When asked for the place of the dog, you can't answer "nach rechts". You have to say "rechts". But when asked for the next direction to turn, then "rechts" would be wrong. It must be "nach rechts".
Both German words, "wo" and "wohin" can be translated as "where" in Englisch, which makes it hard to decide which one to use in German if you are an English native speaker. But sometimes you can use "where to" instead of simple "where", and when ever you can add this "to", then you are asking for a direction, not for a place. And in this case you have to add "nach" in German.
It depends on the context.
"rechts" basically means the opposite of left and is the exact translation of "right". Usually you ask with "wo":
Wo steht das Buch (rechts oder links)?
It's not a about a direction, it's rather about a side or a place.
"nach rechts" is usually used in context of ways and direction. It's rather translated with "to the right". You ask with "wohin".
Wohin müssen wir abiegen? / Wohin hast du das Buch gestellt?