Martin Perhiniak clearly explains about Puppet Warp vs Liquify Filter on his “Yes I’m a Designer” Youtube Channel.
What he says:-
One of the most underestimated tools in Photoshop has to be the Puppet Warp. I Hardly see it mentioned in tutorials and yet I use it pretty much all the time in all kinds of workflows. I know that most Photoshop users would instead rely on the Liquify Filter, so I decided to compare them to each other and show why I think they both deserve your attention.