"Carefully" is an adverb, meaning with care or attention.
"More carefully" is the comparative form of "carefully," meaning with greater care or attention.
Examples:
* "Please handle the vase carefully." (This is a simple statement, implying care is needed.)
* "She opened the box more carefully this time, after spilling the contents last time." (This indicates an increase in care compared to a previous action.)
In short:
* Use "carefully" when you're describing a single instance of doing something with care.
* Use "more carefully" when you're comparing two actions, one being done with more care than the other.