Why is carbon classified as a nonmetal and lead classified as a metal, even though they're in the same group?

Because carbon usually forms covalent bonds, while lead likes to form ionic and metallic bonds (lead's electrons are less tightly bound, so they can be conducting electrons as is characteristic of metals).

Lead is a weak metal while carbon is a non-metal/metalloid

The properties of metals are as follows:

Electro-positive

Electrovalent Bonds

Reducing Action

Lead is electropositive and form cations (+ve ions)

Lead acts as a reducing agent and donates electrons rather than accept them.

It forms Electrovalent bonds.

Carbon on the other hand forms Co-valent bonds. It forms negative ions (Carbide)

Carbon has a few metallic properties (lustre, electrical conductivity, formation of few ionic salts) so it is sometimes classified as a metalloid.