An approximation is similar to an estimate, but they usually indicate different methods of obtaining them.
An approximation is usually obtained by just a rounding (up or down) of a more accurate answer (obtained by calculation); while an estimation is usually obtained by making adjustments to the components of a calculation to make a calculation easier to work out.
Example: 593 divided by 123.
Estimation: 593 is close to 600 and 123 is close to 120, so 600/120=5. The estimate is 5.
Approximation: 593/123=4.8211333..., so 593/123=4.8 approximately (accurate to one place of decimals).
An estimation before carrying out a calculation gives you an idea of what to expect when you do the actual calculation. It's especially useful when working with decimals; it helps to decide where to place the decimal point in the answer.