Pixels control actual available image detail, wether they're stored with a mathematical algorythm that eventually converts to pixels, or stored on a per-pixel basis. DPI applies to the detail with which your printer prints. 
You could print 4 pixels at 600dpi on a sheet of A4 paper, and it'll simply print a representation of 4 huge pixels using 600 drops of ink per inch to do so, but it wont interpolate the 4 pixels or anything. The result will just be 4 big squares, each representing a pixel.

Shrinking an image with many pixels to a small piece of paper doesnt increase the DPI, it just changes the amount of dots used to paint one pixel onto the paper (=dots per pixel). The DPI is a quality setting of your printer. 
More dots = better print quality = the ability to 'paint' more pixels onto paper = the reason why most professional print artists render many pixels.

Generally, you could state that the theoretic ultimate level of detail you could render and print would be to have a pixel for each microscopic drop of ink. The more DPI, the more pixels you can theoretically represent accurately. More pixels then dots, is useless.

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