First of all, I would like to commend you for your in-depth questions. Keep on.
The two words are synonyms, of course. As I see it, they can both express the beauty of the appearance of someone or something. However, I think ωραίος is a deeper and broader word, indicating something more that just good looks. Furthermore, there are some expressions that use only the one or the other, regardless of meaning.
Roughly, όμορφος is more like beautiful, good-looking, handsome, pretty, while ωραίος is more like nice, fine, pretty.
- The woman is beautiful/good-looking = Η γυναίκα είναι όμορφη
- He is handsome/good-looking = Είναι όμορφος
- The food is nice = Το φαγητό είναι ωραίο
- The fair sex = Το ωραίο φύλο
- A nice day = Μια ωραία μέρα
- Beautiful day = Όμορφη μέρα
It depends on what was said. If I say the color you painted your house is 'nice' that might work. But if I say that "The girl is nice."...when the Greek says "όμορφη" which describes "physical beauty" "nice" is not sufficient.
USE THESE INSTRUCTIONS TO BE SURE YOU ALWAYS HAVE THE CORRECT TRANSLATION
1 Always read the comments before posting. Check the heading on the page to see the sentence and its translation. Click on any blue words for more definitions.
2 Read the Tips & notes, on the first page of each lesson you’ll see TIPS. Click on that.
3 Use the Drop-Down hints. Pass your cursor over a word and a list of translations will appear.
The top word/phrase is the best choice.
The other hints will appear at the top when needed. Using the Drop Down Hints will assure that you avoid mistakes and become familiar with the right translation. It's the secret to success. Use the hints as often as you need them.
4 If your translation is rejected you should carefully compare what you wrote with the answer given. If you do not see a mistake use the Report options at the foot of the exercise page.