Rusted days - a WolframAlfa port
I can't see any issues in terms of making your code idiomatic; it looks very Rust-like! I did however notice a problem with your code's logic.
Integer overflow
I tried inputting 31/12/2019 and obtained a panic when I ran it today (07/11/2020). This was because Utc::now().month() is a u32 equal to 11, and the code attempts to subtract 12:
let months = Utc::now().month() - date.month();

This leads to an overflow because we're using unsigned integers. The same story applies for the day, which is also a u32, but not the year, which is an i32 and so functions fine.
Even if we convert the relevant types to signed integers, the following isn't what you mean:
let years = Utc::now().year() - date.year();

This would tell you that 31/12/2019 and 1/1/2020 are a year apart, when in fact this isn't true. Even dividing the number of days by 365 isn't quite right because of leap years.
Making the date logic work correctly is a lot more frustrating than it sounds: the number of months passed depends on the length of each month as well as the Duration instance. That's why Duration doesn't implement anything beyond num_weeks(): it's simply a difference between two dates, so it doesn't know how to account for the varying month lengths.