If I am wrong in this please correct me, but this is how I understand things to be using an example of the popular music industry.
A liberal feminist approach would be to bring in legislation banning misogynist lyrics and videos which objectified women - this would likely have a more immediate effect and would definitely curtail the amount of offensive material which is around on radio, TV etc. However, artists could still be misogynistic etc they just wouldn't be allowed to release music which expressed this. There may still be an "underground" movement of music/videos which are objectifying and sexist. One side effect of having non-sexist music around in the mainstream though might be that it would normalise less sexist thoughts and attitudes, which would of course be a positive thing.
The radical approach would be to make more of an effort to listen to and buy music made by women, to raise its place in the charts; perhaps to set up or be part of a female-positive music label, magazine, studio or radio station; to create music (as a woman) which is empowering, etc, and the intended overall effect of this would be to raise the profile of music which doesn't conform to a sexist stereotype and so reduce the power of the sexist/misogynist/objectifying music until it becomes little more than a niche (for douchebags 
) - of course the downside of this kind of approach is that it may never happen, and/or that it's a lot slower, but if change is happening all over society then we may see a kind of snowball effect where attitudes are changing and actions are changing as a result of this, so it picks up and actually people who never would have identified with being a feminist, let alone a radical one, think "I prefer this music because it doesn't have all that sexist crap in".
I think the difficulty of the discussion in general (not my little example) is that surely it's impossible to state how something as massive as "tearing down the patriarchy" would work in principle. What we can do is share theories on how to "bring down" each smaller obstacle in turn, and that's where the radical vs liberal approach comes in, I think. For me it's important to be thorough and to aim to change thoughts and attitudes first and hope the actions come about as a result of that, rather than forcing actions to change and hoping attitudes change as a result of that, or holding the view that it's actions which are important and an individual's thoughts are irrelevant to the issue. That is how I understand the difference, anyway, but I do admit I find radical feminist theory hard to follow at times or difficult to see how certain actions will have an effect. Perhaps it's all about the snowball!