What is celebrity? What makes it different to fame? Obviously, the two are linked. When you are famous for something – such as acting, or singing – you are expected to be a celebrity as well, to publicise your personal self as well as your public work. How much you see this as a problem, and how much you see it as an opportunity, will have some bearing on whether you're viewed as "famous", or as that more collusive, confected and shallow human commodity, "a celebrity".

Certainly, "celebrity", even though it's craved by many, has negative connotations. Celebrity is widely understood to be more chimerical than fame, less respectable and less trustworthy. Selling coverage of your wedding or photographs of your baby to the media, for example, is seen as broadly understandable – money being money after all – but also greedy and crass, a bit desperate.

Journalists often argue that a celebrity is a person who volunteers to bring their private life into the public domain because a benefit is perceived. Under that unwritten contract, they cannot go bleating about their privacy being invaded when their "real" private life contradicts the idea that they and the media have hawked. If the lack of outrage at evidence that the phones of "celebrities" were hacked is anything to go by, the public appears to agree.

Essentially, I suppose, fame comes from people knowing about what you have achieved, while celebrity comes from people knowing about who you are. Or who you purport to be, anyway. Of the fact that celebrity can be a false flag, a carefully constructed facade hiding the darkest of motivations, there can be no doubt. We have Jimmy Savile to thank for the most horrible possible confirmation of that.

It's plain now that Savile exploited his fame as an entertainer ruthlessly, crafting an image as an indefatigable charity worker in order to gain unfettered access to vulnerable people and abuse them. The fearful symmetry of his technique is dumbfounding. The more he projected himself as a good guy, straightforward, with nothing to hide, the more seemingly isolated his ever more numerous victims became. One imagines that was the point.

Sexual abuse is always about power. Savile maintained absolute power over all of his victims – men, women, boys and girls – all of the time. My feeling is that much, if not all, of Savile's gratification came not from the sexual attacks themselves, but from their continuing confirmation that he was a national puppetmaster. His victims were his trophies. He touched to revel in his own untouchability.

Even the victims courageous enough to think they could tell the authorities, the police or the media, were quickly disabused of their illusion, because in a sense those institutions were Savile's victims, too. They are not his important victims, in terms of personal suffering caused. But they are significant because they had the ability to stop Savile, but enabled him instead. They are important too because they have most to learn.

It's hard, though, to know quite what the lessons are. There's already too much suspicion that anyone who works with young people, or any celebrity who does charity work, has some hidden agenda. The knowledge that something isn't right, but that what's not right hasn't been pinpointed, leads to high levels of cynicism. That, in turn, can be damagingl. People are cynical about celebrity, even as they remain in thrall to it.

Lesson one, for example, seems obvious to many: all accusations of sexual abuse have to be taken more seriously. But even that is not straightforward. There's a sense in which the seriousness with which accusations are taken, is part of the problem. The example of Savile certainly encouraged people to come forward to say they had been abused by others in the public eye. But acquittals, as in the case of Bill Roache, or arrests that led nowhere, such as Jim Davidson's, have prompted some to suggest that it's "too easy" to smear a famous person.

The famous and the celebrated are seen by some as more vulnerable, more likely to attract malicious allegations than others, and more likely to be damaged by the publicity surrounding such allegations, whether they are charged or not. It's weird that post-Savile, a case in which a handful of publicised accusations brought forth a tsunami of others, the possibility of granting anonymity to those accused of sexual crimes as well as those alleging them should still be so strongly on the agenda.

But what's most strange of all is that nobody ever seems to suggest that one way of tackling this problem would be for celebrities themselves to be a bit more sensible. Savile may have illustrated the power of celebrity in the most negative possible way. But his case also reveals the advantage over other people it creates. It's already established that those whose professional authority puts them at an advantage over people – doctors, teachers, bosses – should not use that advantage sexually. Fame and celebrity put people at an advantage too, and yet it is seen as a perk, even admirable, if that advantage is sexually exploited.

Again and again, we hear people's sexual incontinence being defended because "girls were throwing themselves at them". Poor old footballers can't even walk into some clubs without someone trying to pick them up. Maybe, if a sportsman or a comedian sees celebrity as an opportunity to have endless meaningless sexual encounters, we should view it as exploitative and pathetic rather than lucky and glamorous. University lecturers, after all, were once considered to have similar licence, but are now advised to keep the door open when they are alone with a student.

After Savile's crimes were revealed, it became apparent that people did know about some of his proclivities, but brushed them aside, sometimes citing "groupie culture" as having been acceptable in the past. What this really says is that celebrities were expected to be sexually exploitative, and that people therefore found it hard to know when someone was stepping over the line. In these more enlightened days, it's understood that people under 16, and people who do not give consent, are over the line. It's not that enlightened though, is it?

Instead of talking about changing the law to protect vulnerable celebrities, or huffing and puffing about how evil accusers conspired to get some innocent celebrity into the dock, why not make it clear that celebrities, like lecturers, need to keep the door open, that even if they did nothing legally wrong, people in their position need to take more care?

It is not unreasonable for employers, the police, the media and judges – instead of agreeing that someone was subjected to a dreadful ordeal and isn't that a shame – to say that people in the public eye are foolish and irresponsible to allow themselves to get into situations in which accusations can be made.

Fame doesn't make people vulnerable. It makes them powerful. If they choose to use that power to get things – and people – that they might not otherwise have, even if they are doing nothing legally wrong, then that's their choice, but it's not a noble one. It's about time our institutions found the clarity of mind to say so. Simply telling Savile that there had been complaints, and that he should ensure that he was no longer ever alone with others might not have stopped him, but it is the least that should have been done.

Few people in the world are as calculating as Savile, thank goodness. But part of the reason why he got away with what he did, especially early on, is that celebrity confers further freedom on people, when it should demand greater self-discipline. Maybe, instead of enjoying the spectacle of a celebrity being cut down to size, our culture needs to let them know that it is incumbent upon them to genuinely try to be bigger.