For example, if you're having doubts about a relationship or feel concern about a sensitive subject to the other person, you can write your thoughts to work it out on a separate piece of paper to securely dispose of after, or (similar to a suggestion elsewhere in the comments) a text file that isn't saved or gets deleted after.
Some may argue for great openness about thoughts in a relationship, but I think it's often useful to work things out on your own to clarify your thoughts and position, before a crucial discussion.
The main benefits include the flexibility to easily draw and write formulae if thinking about a technical topic, the ability to recall what you write easier than what you type, and enjoyment of the feeling of writing on paper. The feeling is actually really nice, and I believe it even inspired essays by Japanese authors about the feeling. The Guardian interviewed some researchers about the differences at: https://theguardian.com/science/2014/dec/16/cognitive-benefi...
The main drawback is the ability to search, though I personally rarely look back at past entries (the main benefit is during the act of writing). When brainstorming about something work-related, though, I do scan the related pages with my phone. Digital handwriting (e.g. Surface or iPad) with an app that does hand recognition is an alternative, though the feeling of writing with a good pen on quality paper is much nicer than a stylus on the display (even with screen protectors that simulate paper).
If you’d like to try, you can get an excellent setup for less than $50. You can get a nice pen (Pentel Energel, Zebra Sarasa, or Uniball Signo) and notebook (Rhodia, Midori, or Leuchtturm1917) and try it out for a few weeks. r/pens and r/notebooks talk about the differences, and also suggest other alternatives for writing tools.