でしょう has more uses aside from its overlapping meaning with ね。 でしょう can also mean like “perhaps” in conjecture statements and is used in weather forecasts for example. It can also be used to make questions more polite and indirect in teineigo or keigo (polite language), but in casual speech it tends to be more forceful than ね。
Compare casual speech:
今日は暑いね “It’s warm today huh?” (seeking agreement)
今日は暑いでしょう？ “Don’t you think it’s hot? (What do you mean it’s not hot?)” (assertive)
明日は大丈夫でしょうか？ “Would tomorrow be okay, perhaps?” (seeking agreement)
明日は晴れるでしょう。“It should be a clear day tomorrow.” (conjecture)
Perhaps another way to look at it is that でしょう is a more feminine or polite version of だろう, whereas ね is a more simple agreement-seeking particle.