Some problems with independence are things like the remaining role of the Lord Chancellor (still appoints some judges, can accept or reject proposals of the JAC and his responsibility for day-to-day running of the Courts), the choice of people to be members of the JAC, the influence of the media (it may be seen as good for the media to encourage judgements by judges to reflect public opnion, but this can go too far). In some cases the media has even drawn politicians into getting involved, for example when Michael Howard got involved in the sentencing of the killers of James Bulger.
Judges are trained in order to be politically neutral, and must always keep everything out of their minds except the evidence presented. Neutrality is often criticised, as most judges are white, middle-to-upper class with a private school education and graduates from Oxbridge. They tend to favour the establishment (the state) over the rights of the individual in most cases and some think that they hold a conservative bias. Judges in general try to protect the natural order and hierachy rather that protect the rights of the "little man", for example in the case of accused terrorists like the "Birmingham Six" and "Guildford Four" where verdicts were later overturned as the evidence was improperly collected and perhaps falsified - this was obvious at the time of the judgements, but the legal officials favoured the establishment (in the form of the police) over obvious justice. Judgements also tend to be against those who are at odds with the beliefs of the judiciary and again they tend to follow the government view of the time - for example, when Thatcher was on her crusade against the unions, the judges also tended to rule against them even when it seemed not to be fair to do so and was affecting their right to strike and protest.
Some feel that these arguments against neutrality are less valid today than before. The judiciary is becoming more and more diverse with more female, less priveleged and non-white judges, and judges have shown more willingless to challenge the government eg. challenging Michael Howard's direct control over sentencing as ultra vires when he was Home Secretary in the early 90s and the Supreme Court quashing control orders from the Treasury in 2009. Some even think that the judiciary has a more liberal bias now.
Hope that helps.