Both are vector graphics (typically) and both can be imported painlessly into a pdflatex document (so let's say we ignore dvi for this question).
What are the advantages and disadvantages of each? What should I use?
Use PDF. EPS cannot be imported directly by pdftex but must be converted using something like
epstopdf. These conversion procedures will often cause unwanted changes to the graphics, such as lossy JPEG encoding of embedded bitmap images. Pdftex will include PDF files directly without making any changes (except for unifying fonts, and even that can be disabled if needed), so you can have complete control over the final result by generating a PDF which is exactly as you want it (assuming your image editing software gives you control over image encoding lossiness, colour spaces, etc).