In practice, for stock-options, volatility is strike-dependent ie implied volatility has a skew (lower strikes have higher implied volatility - the leverage effect). Also, the volatility is dependent on time to expiry of the option.
Also, as I've mentioned in the HN thread for your previous article, single-name stock options which are exchange-traded
have an American exercise type. The valuation of these cannot be done using Black-Scholes if they are (a) long-dated (ie time to expiry is quite long) or (b) they are deeply in the money/out of money or (c) have an underlying which has a significant dividend yield (in which case you'd want to own the stock rather than the option).
Puts behave different than calls if the exercise-style is American (puts have a limited upside - so you wouldn't wait too long if the underlying stock has fallen far enough - your payoff is not likely to be larger).
You may want to discuss this and option greeks the next time.