emc summed it up really well.
i would add that these terms don´t have any fixed meaning, they´re variably used by members of subcultures to refer to themselves, at times are epithets, can refer to fashion or lifestyle or beliefs...
to most people they evoke mainly a certain set of aesthetics, and what comes to most peoples´mind as "hippie" are aesthetics commercialized by mary quant shortly after, if not concurrently of the hippie movement becoming widespread, and the same exact thing happened in punk with malcolm mclaren and vivienne westwood
in both cases, the name was first used in contexts that i don´t think apply anymore, sure you can appreciate "hippie" values and practices or "punk" music and fashion, but the levels of identification to these terms more than 30 years after they came into use for tv culture to present youth culture in an understandable way to the masses, is ridiculous.
at the essence of hippie is - or was orignally - a focus on communal ways of living, sharing, being more than the ego, and being inspired by forms of life, art and resistence everywhere.
meanwhile punk was always supposed to be about rejecting dominant aesthetics and rigid ways of thinking and living, courage to be completely individual in your personal life and public interactions, and expressing discontent (that hippies from the previous generation had, but also still had optimism) with the state of the world with no restraints
both original subcultures also practiced innovative forms of political resistance, but it seems that people who identify themselves with those subcultures today aren´t very concerned with continuing the innovation, punk and hippie apparently both come with a solidified set of activities, and meanwhile activists who are more interested in innovating techniques for direct action or community work aren´t allying themselves with any cultural movement
most of the time when you i meet someone who is quick to talk about how "punk" or "hippie" they are, they´re sheltering in a prepackaged revolt that soothes their discontent with their own life, makes them feel good for being part of a group, whatever, but has little to do with addressing the things that both of these movements set out to address
sorry for writing an essay and i am aware that because i have patches on my clothes in my profile picture, it would be handy to label me a punk at first glance, i could care less what i get called but i think it´s sad or at least boring when people identify themselves as an example of a certain preconceived kind of human