Here are the tradeoffs as I see them:
1. The piston/propeller missile is slower, so it's easier to shoot down. If you have a certain detection range, you have more time between detection and impact. The traditional AEGIS type air defense on a high end ship has three layers -- long range missiles, short range missiles, and guns. Guns are the cheapest with the most ammo, but they're short range, so with a fast missile you only have a matter of seconds to shoot it down. If there are a lot coming at you simultaneously, you need more guns to get them all. If they're moving slower, this is a lot easier.
2. Lift is proportional to the square of the airspeed, so the slower you go, the bigger the wing you need. Compare for example the LRASM (the latest U.S. anti-ship missile), to a Cessna 172. They each weigh about the same (2500 lbs), and have very roughly comparable payload and range. Since the Cessna goes about 120 knots and the LRASM goes roughly 600 knots, that's a 5x difference in speed, so the Cessna needs roughly 25x bigger wings to stay in the air. You see this with the span, the LRASM has a span of 8 feet and the Cessna is 39 feet. Obviously the chord on the Cessna is a lot bigger too. The bigger wing will give you a bigger radar cross section, so you're detected sooner. It also makes the missile cost more due to all of the structure, and it makes it hard to put it in a launch tube if the wings are supposed to fold up.
I'm sure there are a lot of creative solutions to this with flexible wings, etc, but there are good reasons to prefer a faster missile. Whether that ends up being the war winning solution though, time will tell.
Things are a lot different than 20-30 years ago in terms of tradeoffs though. Back then your guidance system with a radar, computer, radio/satellite transceiver, and inertial navigation would weigh over 100 lbs, and cost over $100k in today's dollars, so making smaller missiles made less sense. Now, you can do guidance with GPS, strapdown inertial, and optical, and there are low power and low cost radio transcievers. Instead of a radar, you can do the terminal guidance optically with a multi-teraflop GPU that costs $50.