You said "I'd like to show the importance of this scene." The word you used was "importance," so use the word "important." It's as simple as that. A writer's job is always to communicate well. If you know that word X has the right meaning and you're not quite sure what word Y means, use word X. If you communicate your meaning well, nobody is ever going to give you a bad grade for doing it in plain words instead of fancy ones. In fact, most guides to good writing style will tell you to use the plain words. Here is an example illustrating the meanings of the words in English. "Sherlock Holmes, can you deduce who is stealing cookies from the cookie jar." "You know my methods, Watson! Apply them." "Ann is wearing red shoes." "Irrelevant, Watson." "Barbara loves cookies." "Better, Watson, better! Relevant, but unimportant. Who doesn't love cookies?" "Cindy was seen in the kitchen with her hand in the cookie jar and cookie crumbs all over her face." "Now that, Watson, is an important observation."

"relevant" is a horrible "faux ami" in Romance languages vis-a-vis English. Students often use "relevant" to mean "important". "relevant" means "in question" or "pertinent". Sometimes a relevant thing can also be important, and you have to decide if you want to refer to its importance with "important" or its pertinence with "relevant".

Can you use both words? Obviously, the scene is important in the sense that you couldn't leave it out and have the play remain the same. In other words, it's not a minor detail ... the scene is essential to the progress of the play, as you correctly pointed out. Relevant means related to the topic at hand; so, for example, tests often ask you about relevant ideas or details. If we discussed the balcony scene, we might say that Romeo's portrayal of Juliet as the sun is relevant to the development of the romance, while Juliet's dress is irrelevant. For a fully nuanced commentary, it seems to me that you could discuss both the importance of this scene and various elements and details that are relevant to the reasons it's important (those which you already mentioned ... romance, personality, plot development).

Relevant = How well can you relate to it? For example, something most people can understand because they went through something similar. Important = It is a key factor for the play to progress (as you mentioned). So, we would use "Important".