Are there any difficulties or benefits of having judges appointed for an indefinite term?
Yes, there are both difficulties and benefits to lifetime appointments for the judiciary.
The main reason our federal judiciary system was designed this way was to ensure that judges could remain independent of the will of Congress, the President and the popular vote. It is much easier to protect democratic rights for a minority, even an especially unpopular one, if the courts and judges do not need to worry about whether or not they will be re-appointed/re-elected. This allows us to protect constitutional principles and guarantees even when it is inconvenient or unpopular.
The lifetime appointment also allows judges to gain experience over time becoming, essentially, constitutional scholars who know both the law and legal case history. This makes it more likely court proceedings will be fair and legal.
On the other hand, in order for a judge to be removed, they must break the law in a way that can be considered an impeachable offense. So their decisions may be controversial, or legally flawed, and there is little the plaintiffs or defendants can do about it except appeal the ruling up the chain and hope for a more favorable or appropriate ruling from higher courts. The Supreme Court's ruling, even if flawed, is the last word. It concentrates a great deal of power in a small number of judges to interpret the Constitution for us, which is a weighty responsibility.