Purposely means on purpose. It is a synonym of intentionally. When you mean to do something, you do it purposely.
Purposefully means (1) with a sense of purpose, or (2) with determination. For example, when you are determined to ask your boss for a raise, you might walk into her office purposefully.
These adverbs are often understandably confused, and they are more or less interchangeable in some uses, especially where actions are both intentional and determined. For example, a football player who intentionally fouls another player does so purposely, but we can also say he does so purposefully if he does it with a sense of purpose or in a determined way. In uses such as this, the distinction can be blurred. Purposely is the broader and more common term, so it’s usually the safer choice.
It has also been claimed Natalie purposely forgot to include Sarah in her Oscar acceptance speech. [Stuff.co.nz]
I am being purposely vague with the details because you just have to read the delicious punch and counter-punch over at Eater. [Los Angeles Times]
There’s even the suggestion that telecommunications companies overcharge purposely so that they can offer a discount. [Globe and Mail]
Schoolchildren with a homemade banner march purposefully past a popcorn stand followed by a chain of exploding firecrackers. [Guardian]
He snapped open the shutter and then walked in front of the camera—in the shot—and strode purposefully around the scene. [New York Times]
Cora. purposefully took extra grounders at third during the spring. [Washington Times]