There have been a lot of studies on handwriting and memory. The Wall Street Journal article How Handwriting Trains the Brain talks about one such study that shows that children learn to recognize letter shapes faster and more accurately if they write them out rather than typing them. Writing by hand also helps adults who are learning a graphically different language, "such as Mandarin, or symbol systems for mathematics, music and chemistry." The article suggests that it's the physical activity of writing and the link between the hand and the brain (not the extra time taken) that helps improve learning.
The Telegraph article Write it don't type it if you want knowledge to stick mentions that the increased time that it takes to write things out by hand may play a part in increased learning, but it also talks about the increased mental effort and feedback that the brain receives from your hand when writing vs. typing.
The LifeHacker article Why You Learn More Effectively By Writing Than Typing talks about the link between writing and goal achievement.
A couple of studies, though, substantiate why the physical act of writing really does boost learning and goal achievement. Hoping to provide actual scientific proof on the efficacy of writing down and sharing goals (to make up for an often-quoted mythical Harvard/Yale study of goals), a psych professor at Dominican University of California found that people who wrote down their goals, shared them with others, and maintained accountability for their goals were 33 per cent more likely to achieve them, versus those who just formulated goals. (One can argue that in this instance, typing would be equally effective; see “Why Writing Works Better Than Typing” below for why writing still may be better.) Another study found positive effects of writing on learning foreign words, and a survey of note-taking studies found several examples where taking notes helped students with recall and academic performance.