What would be the difference in a coil of wire and just a cylinder of solid copper?
The part you are missing is that what looks like uninsulated wire actually isn't.  A lot of enamel coated or "magnet wire" can look like bare copper at first glance, but the wire is actually coated with a thin semi-transparent insulation layer.  The reason for using thin insulation is so that lots of turns of the coil can fit into the tightest possible space.

Electrically, a coil of wire is quite different from a cylinder.  To make a magnetic field, you need current flowing around where you want the field.  Think of a cylinder of current surrounding the area.  The magnetic field is proportional to the total current in the cylinder.

A coil is sortof a cheat on that.  The same current is re-used each turn to add to the apparent cylinder current.  Let's say you have a coil with 100 turns.  If you run 1 A thru it, each of those turns contributes 1 A to the overall cylinder current.  You get the same magnetic field as if it were a solid cylinder with 100 A running around it.