How is multiplying exponents (x² ⋅ x⁴) different from raising a power to a power ((x²) ⁴) ?

When you're multiplying exponents such as your example, x^2 * x^4, you would add the exponents. The answer of the example would be x^6.

When raising a power to a power, you would multiply the exponents. The answer to your second example would be x^8.