A european option can only be excersized on a pre-defined expiry date, while an american option can be excersized at any time before the expiry date. Since an american option gives additional rights to its holder, it can never be cheaper than a european option with the same characteristics.
You can see exactly how much price difference can there be in this study, which consists of a long position in an american put and a short position in a european put option of the same strike and expiry. Wherever there is no difference, the value of this portfolio is zero, and for the points where the american option is more expensive this portfolio has some positive value. When the risk free rate goes to zero and in the absense of dividend payments by the underlier, european and american prices do not diverge, you can see that as you change the rate towards zero, the graph turns into a zero-line. You can also study the difference in delta and other greeks by swithcing to corresponding graphs.
You will probably notice, that graphs in this study are not always smooth and have visible steps. This is an artifact of the binomial model, which is used for american options.
Want to learn more? Download now an interactive reference application for iPhone.
The screenshot shows the following portfolio:
American put struck at 10.000 with expiry in 30 days
European put struck at 10.000 with expiry in 30 days
This is an excerpt from iOptioneer option trading reference application. In order to build the real-time dynamic strategy graph and run simulations you will need to download the application from App Store.