Arduino Mega vs servo shield
The idea of that particular servo shield is that it provides 16 PWM (pulse width modulation) channels. These are all operated simultaneously with no additional processing overhead on the Arduino. 

The Arduino Uno itself does not offer 16 (or 18) hardware PWM channels but only 6 (see here). So driving 18 servos without an additional shield would require software PWM which requires some of the Arduino's ressources to be devoted doing the calculations. This may or may not interfere with the rest of your program.

Furthermore the Uno does only provide 14 dedicated digital GPIO pins in total (+6 multiplexed digital/analog pins, as ZS6JMB's answer points out), so additional hardware would* be required either way - or using an Arduino with more pins (such as the suggested Mega). The Mega however offers only 15 hardware PWM channels, leaving you with soft PWM again. (* assuming some pins will be needed for sensors or user interfacing).

An alternative is this board or, uhm, well, cheaper clones if you like. It features the same chip as the shield but with a PCB half the size and thus skipping the "free use area". I expect it to behave the same way like the shield and operate with the same libraries as the both the shield and this board communicate only via I2C. Both can be chained (or stacked for the shield if you will) for up to 992 PWM outputs (62 boards).