The value of an American call option is closely related to that of a similar European call option. Provided that there are no dividends paid on the underlying between now and expiry, the existence of option value will ensure that the option is worth more than its intrinsic value. Therefore investors will always prefer to hold an America option to expiry.
The above means that an American call will always generate the same cash flows as a European call option with the same underlying, strike price and expiry, and is therefore worth the same.
If there are dividend payments between now and expiry, then it can be shown that the optimal time to exercise is just after the underlying goes ex-dividend. This somewhat simplifies valuation.
There is no such simplifying relationship for the value of an American put option.
This does not mean that American puts cannot be valued! It does mean that different, and, often, more complex approaches must be used. This usually means using computational approaches such as Monte-Carlo simulation, rather than a formula like Black-Scholes.