The key to weightlifting/powerlifting shoes is the hard sole, as it helps with leg drive out of hole in certain lifts. As for the heel, it is more of a lifter's personal choice in powerlifting, you'll notice over the years that heel height, etc.... changes a bit over the years.
For weightlifting, it's the combo of the heel plus the hard sole that is helpful during the lifts. It has to do with leg drive and ankle, shin, and pelvis position during certain parts of the lifts. It's a bit technical for a short post, but you'll notice it and understand it if you ever take up O/Ling. Most Weightlifters prefer the low cut shoes for ankle mobility.
Also, needless to say, your lifting shoes should be the tightest you can get away with for stability, perticularly in width.
For most general lifting, Adidas Sambas or Converse Chuck Taylor's should be good enough.
