The plastic clips won't gouge up your shoes as much, so that's sort of a plus, but they definitely flex more so when you really start hammering, your feet with float all over the pedals.
Good pedals are a fine investment, as they are one of the three points where you actually interface with the bike (the others being the bar and saddle.) If your pedals are crap, the whole bike can feel like crap.
With regard to terminology:
The quill is the 'H'-shaped piece that actually screws into the crankarm.
The cage is the plate or plates that are bolted to either side of the 'H' to create the platform.
The clip is the piece (sold separately) that bolts to the front of the pedal that loops up and over your foot.
The strap is a piece of leather that is threaded through the pedal and the cage and loops around your foot.
The cage gives your foot a place to rest on, the clip keeps the foot from sliding forwards and backwards, the strap ensures that the foot is securely attached to the pedal. Riding with clips but without straps, or the straps overly loose is essentially riding without clips at all.