"I dance better than you."
Translation:मैं आपसे बेहतर नाचता हूँ।
Yes. They are basically the same.
Tujhse is derived from Tu. Example
Tu ja raha hai - You are going.
Tujhe(तुझे) kya hua - What happened to you?
Peter ko tujhse baat karni hai - Peter wants to talk to you.
All these have an equivalent in tum and its forms.
Tu =Tum. Tujhe=tumhe(तुम्हे). Tujhse=tumse
Tum ja rahe ho - You are going.
Tumhe kya hua - What happened to you?
Peter ko tumse baat karni hai - Peter wants to talk to you.
Tu and its form are considered rude in most हिंदी speaking states. I don't know why duo decided to teach it as default.
Always use आप aap. And otherwise use Tum/tumse/tumhe.