Should I use reference manager software?
You do not need to use reference managers. However, if you use some reference managers like Mendeley then, some of the benefits are obvious:


The pdf of your article is linked to the reference itself.
You can organize references into logical groups (areas, subjects, disciplines, sub-disciplines) by utilizing folders and tags.
You can one-click save articles and organize them from your browser. This is invaluable in saving time on organizing.
You can sync across multiple devices like desktops, laptops and tablets. For instance, Mendeley has an iOS app which does this for you. I use Android so I workaround using Droideley or Scholarley.
You can read your articles inside the reference manager and then do useful things like highlighting and annotations which can then be saved as notes for future reference.
You can export to specific citation styles. This is very useful for someone doing interdisciplinary work. For instance, I have to constantly use the same or similar bibliography using APA style, ACM style, IEEE style.
They are very useful for multiple collaborative platforms. For instance, when collaborating with some computer science colleagues, I generally use LaTeX and Mendeley does a brilliant one-click job of exporting to BibTeX. When I work with social scientists, I generally use MS Word and there is a MS Word plugin for Mendeley which dynamically adds references in-line as well as the full bibliography at the end of the document.


I hope this has been partially useful to you. This is a little Mendeley focused and I am in no certain terms claiming that it is the best or the most useful. However, for me, it certainly has its advantages.