The basics of the legal system aren't all that different. The UK is built on English "common law," which is basically the collective term for the rights, procedures, and case law precedents developed since the Magna Carta. Most former English colonies including the US follow the common law tradition, meaning that the basis of US law is English common law as it existed at the time of the American Revolution.
Common law principles included such things as jury trials, impartial judges, protection of property rights, and more technical concepts like "res ipsa loquitur," the Rule Against Perpetuities, and the Statute of Frauds.
Of course in the 200+ years since, significant differences have developed between US and UK law. For example, the UK has abolished the right to jury trials in civil cases, but in the US that is still constitutionally protected. Another often cited example is in the treatment of riparian rights to a common source of freshwater. The English rule is first in time, first in right, while the American rule is reasonable use. Still another example is in the award of attorney's fees after litigation in court. In the US, each party pays its own fees unless otherwise directed by the court. In the UK, the losing party pays both its own and the winner's fees.
Because there is still a lot of overlap between US and UK common law, particularly on very old legal concepts such as contracts and wills, US law schools still teach many old English cases from the 17th, 18th, and even 19th centuries in addition to US cases.