In the United States tuition charges vary dramatically between and within universities. This convinces me that our own government will be well-advised to maintain strong control over tuition-fee levels. (Who gets the fees and whether the control should include lab-and-museum trip charges are different matters.)
US experience suggests that when universities are able to charge their own fees they first undertake to scale or discount their fees according to family means. But competitive admissions pressures soon induce them to discount their fees for all sorts of other reasons: to get students with sparkling A-level predictions, women engineers, even the gifted cellist. This will tend to reduce the amount of relief they give to students on the basis of financial need, and mean that some richer students will pay less than some poorer students at the same university. Much of the money spent, or foregone, on competitive discounting will in fact be wasted - cancelled out - as competing institutions make similar offers.
Power to set their own fee levels will also tempt universities to vary tuition fees between departments, accentuating rich-student/poor-student differences between subjects and career fields. This, by and large, does not happen in the US because undergraduates do not "declare their majors" until the end of their second year. British universities would have more scope and incentive to do this.
Professor of North American programmes Sussex University