Do the two have any differences and do both mean to say something unintentionally?
The Cambridge Dictionary gives the default (most common) sense:
- in a way that is not intentional
But Merriam-Webster adds a second sense (for the adjective; the adverbial sense follows):
inadvertent adjective ...
... 2: not focusing the mind on a matter : inattentive
The second sense here is obviously largely synonymous to 'without thinking', though both phrasings are usually used when a person says a certain thing unguardedly rather than when referring to a complete airhead.
The default sense might be considered to be synonymous here, but people usually say what they intended to say a split second before; here, it would be the ramifications that were unintended.
I'd always use 'without thinking' here, unless I let out a secret I wasn't aware was a secret.
There is a difference in my opinion, and its illustrated by these two examples:
To say something without thinking...
A man is walking down a busy street in a crime ridden part of the city and talking on his phone.
"Hey Bill, yeah I'm carrying the 5000 cash in my bag and I'll be there in ten minutes."
He didn't think about the consequences of saying what he did.
To say something inadvertently...
Donald: "I hate paying tax. Last year I didn't pay any tax at all and it felt great."
Donald has inadvertently admitted that last year he either didn't have a job or he ran a business that didn't make any profit.