I can see many advantages using cleat style pedals. An obvious one is that you will not slip and that is especially important in wet weather. Bruised shins are not fun. Also, it will help you to "bunny hop" your bike, jumping over kerbs and obstacles. Better power seems to be a claim that many people make in using cleats and I don't disagree, though 50% better is a bit hard to digest.
My esteem ride buddy Chris of CGOAB fame feels that:
SPD cleats are so efficient, there's no turning back once you get used to it. I've used up more than a dozen and that's spread out over a few bikes. You can push and pull up on the pedals.
Disadvantages can include safety - takes skill to disengage from the pedal during emergencies and spills (I've seen friends falling like pins at traffic stops). Getting stuck with SPD shoes that hamper the way you walk when you get off the bike. Increased cost - getting the shoe/pedal set up can easily add $200 and more.
The freestyle/traditional way is the preferred method for beginner/casual type cyclist. Whatever shoes you wear will work, but a firmer sole is better to transfer the power. Which explains why it remains a favourite among the European commuter cyclists.
No drama pedalling, and easy to jump off the bike if the need arises. Of course, this can be painful when things can slippery. You can forget about jumping your bike and spinning at a high cadence is not as efficient. Also, the lycra brigade will not be caught dead with these "novice" pedals.
Grant Peterson is a strong advocate for pedalling "free" style. He writes:
The most important and liberating thing I've learned in 40 years of riding nearly daily, is that normal shoes and pedaling unconnected is the way to go... and if you're looking for an excuse to head out on a ride in your Hush Puppies, now you have it.
I don't hope to convince you either way but I guess the general rule is if you are looking for that edge in cycling performance, get cleated. If you are just cycling easy to the shops to pick up milk, then platforms rule.
So as an adventure folding bike tourer, what do I prefer? I think the best of both world solution is the Zefal mini toe clip. This is an unbelievable US$8 solution.
I have been touring with this on cycling shoes or Target Crocs sandals with total ease, confidence and comfort. It does the job of keeping my feet from slipping and yet disengages easily when needed.
However, the more important question to ask is which is more your style of riding? Get that answered and settled quickly, then go enjoy quality time on the saddle instead of endless chatting on bike forums. I suspect there is enough room in this big world to accommodate and respect two differing views. Don't you?