"What is the date today?"
这个说法比较正式，适用于工作场合。
例句：
Can you tell me what is the date today?
I need to know what is the date today for my appointment.
"What's the day today?"
这个说法比较口语化，适用于日常生活对话。
例句：
Hey, do you know what's the day today?
I just want to confirm what's the day today before I make plans.