To my mind, a "part" is something so essential to the operation of a bicycle that if you remove it, it's not really a bicycle anymore. Fork, handlebars, pedals, bottom bracket, etc.
"Equipment" sounds to me like additional stuff that you use around or on a bicycle, but aren't quite as essential. May or may not actually be attached. Pump, lights, water bottle, tools, cyclometer.
Not quite sure about things that bolt on (or otherwise attach semi-permanently) but that have debatable essentialness like fenders/mudguards, racks, baskets, bottle cage, etc.
But it does seem like we're a bit inconsistent with how we use these two tags.
Right now equipment has description that doesn't explain anything and covers everything: "Equipment (as opposed to riding or technique)."
So, how should we be using these tags? What should each of their tag wiki excerpts say? Does the answer to these questions mean we should go do some tag cleanup on those questions?
Tag cleanup finished. equipment is marked deprecated, components is a synonym for parts and all relevant tag wiki excerpts updated. Retagged everything that had equipment, sometimes simply removing the tag, other times replacing with parts, accessories and/or tools as appropriate for the question.