Learn German: Hat jemand Draht statt eines Gummibands? - Does someone have wire instead of a rubber band?

The preposition statt (instead of) requires the genitive case. Since das Gummiband is a neuter noun, its indefinite article changes to eines in the genitive, and the noun itself gets an -s added to the end.

Is there a difference between statt and anstatt?

No, they mean exactly the same thing and both require the genitive case. You could just as easily say anstatt eines Gummibands here. Statt is simply the shorter, slightly more common version.

Would it be wrong to say statt einem Gummiband using the dative case?

Strictly speaking in standard written German, statt takes the genitive (eines Gummibands). However, in everyday spoken German, you will frequently hear native speakers use the dative (einem Gummiband) after statt. Learn the genitive for tests, but expect to hear the dative on the street!

Why isn't there an article like einen before Draht?

Draht (wire) is being used here as an uncountable noun, just like in the English translation. When uncountable materials or substances are the direct object of a sentence (in the accusative case), they usually do not take an article in German.