The primary advantage that I notice of writing complex thoughts by hand, for example a first draft of something important, is that insertion of more than a few words is difficult on paper. Therefore I tend to work just a bit more deliberately and focus a bit more on the narrative structure. The results tend to have a more coherent narrative flow, be easier to read, and be more organized than a draft written on a computer where I can jump back and forth between partially-completed paragraphs or sections.

Similarly, decreasing the accessibility of information, code, etc. helps me develop the narrative when making the first draft.

I get a pretty big productivity bump from bringing a notebook to a desk or table away from a computer (with its attendant distractions) and spend an hour or two, producing a page or two, that require much less editing later. I can leave little spaces for numbers or other specifics that I don't have on hand, and make marginal notes for other things to research or fill in later, while focusing on the structure and organizing my thoughts and arguments. Afterwards I can make figures, check calculations and statistics, etc.

I also frequently draw out powerpoint presentations on graph paper, maybe 8 slides per paper page, and just write the basic points and sketch in the figures I want to add. Then I go through and actually make the slides at the computer.