Why can't we add resistance and reactance directly to form impedance?
Because in complex algebra,  magnitude of a complex quantity \$ a + jb \$ is not equal to \$ a + b \$ . But \$ \sqrt {a^2 + b ^2}  \$. 

The resistance is a real quantity. While reactance is an imaginary quantity. 

Resistance = \$ R \$

Inductive reactance = \$jX_L\$

Capacitive reactance = \$ -jX_C\$

The sum of them will be impedance, \$ Z = R + j(X_L - X_C)\$ , which is a complex quantity. 

Why Resistance is a real quantity and Reactance is an imaginary quantity ? The Answer is in their respective phasor diagrams between Voltage and current. 

For eg. in series RLC circuit: