What you're seeing is the
insulation on the old wire, which is white (and red and black on the other wires in the cable), but the
sheathing on the new cable. It's just a confusion of terminology.
wire is a single strand of copper (or aluminum), often coated with a colored
insulation (except in the case of ground wires, which are allowed to be bare). The size of the wire is denoted by its gauge (in the US, at least): #14, for example. The color of the insulation is sometimes an indication of its purpose, although that's not always the case. Black is usually hot, white is usually neutral, red is switched-hot or traveller - but again, these are standards, not rules.
Several wires bundled together form a
cable, which is coated with a colored
sheath. The color of the sheath typically denotes the size of the wires contained within, although that's not always the case (especially near outlets, as the yellow sheath of 12/2 or 12/3 cable often gets sprayed by paint or texture by the drywall crew, making it look white). Cable is named or marked by the size and number of the conductors (excluding ground): 12/2 has 2 #12 wires plus a ground; 14/3 has 3 #14 wires plus a ground.
The best way to know what size the wire is is to measure its cross-sectional area or diameter, or check the sheath of the cable for any markings which indicate the size of the wires within.