Convert the first number to the following

222^{333} = (2\cdot111)^{333} = 2^{333}\cdot 111^{333}=(2^3)^{111}\cdot 111^{222}\cdot 111^{111}

Convert the second number to

333^{222} = (3\cdot111)^{222} = 3^{222}\cdot 111^{222}=(3^2)^{111}\cdot 111^{222}

Divide the first number by the second

\dfrac{222^{333}}{333^{222} } =\dfrac{8^{111}\cdot 111^{222}\cdot 111^{111}}{9^{111}\cdot 111^{222}} =\Big( \dfrac{8}{9} \Big)^{111}\cdot 111^{111}=\Big( \dfrac{888}{9} \Big) ^{111}>1

Since the quotient is greater than 1, the first number is greater than the second one.