Why the current through a conducting wire and a resistor are the same though the resistor has a greater resistance?

When a resistor is connected to a source using conducting wires, both are in series. The current through the wire as well as the resistor is determined by the total resistance of the circuit according to the relation

V = IR

Similarly, when a current is passed through a heating element, the connecting wires are not heated; only the heating element gets heated up since when a constant current is passed, the heat produced is directly proportional to resistance. The current in a series combination cannot change from component to component as current is the rate of flow of charge and the charge supplied by the source must be flowing through all the components connected in series; there is no creation or loss of charge along the circuit.