So the most obvious is that the U.S. Miranda Rights specifically mention right to legal counsel and right to state provided legal counsel (Public Defenders) if you cannot afford legal counsel (Contrary to some opinions, these guys are very good at their job... it's just that they are also very over worked and private industry pays better). While the right exists in the U.K., the U.K. version of the required reading of rights only speaks to right against self-incrimination, which, if you want a difference is a good place to look.
In the self-incrimination clauses, the U.K. and U.S. versions are very different. The U.K. right is a qualified right where as the U.S. is an absolute or unqualified right. This is a distinction which sounds silly upfront but is very serious in how things will transpire.
Suppose that you are arrested for the murder of your spouse. You definitely did not do and the "one armed man" definitely did. Either way, you remain silent during interrogation. At trial, your defense is "It wasn't me it was the one armed man" and you intend to present evidence of this. In the U.S. this would be permitted, no further questions asked (or at the least, defeated by other means unrelated to you giving the cops the cold shoulder). In the U.K., this would be first be challenged by the prosecution with "Why didn't you say this when you were arrested?" and your silence on this matter will be used against you.
In fact, asking that challenge in the U.S. is very inappropriate, as was recently seen in the Kyle Rittenhouse trial, where the Prosecution did ask that up front to Rittenhouse, prompting a scolding from the judge out of view from the jury.
The reason for this is that in the UK there are more strict rules placed on cops during interrogation than there are in the U.S. (In the former, cops cannot lie to you about the facts of the case and they cannot interrupt your statements to them once you start to respond. This is par for the course in the U.S. for cops. In fact, in the U.S., shouting "It was the one armed man" on arrest can do more damage than just shutting up until you're before a judge and jury since that lets the prosecution use the implausibility of a one armed man against you (in both nations, statements that are against your interest do not violate hearsay rules, thus, the cops will only use such a statement against you... it's your job to prove it true or at least plausible enough to make a jury doubt the cops are right.).
Also note that this is England and Wales jurisdictions only. Scotland, having its own legal system, retains the right against self-incrimination as an absolute right.
Also a big obvious one but the read rights would not be called the "Miranda Rights" by the police or legal community (it may be, by the crooks they are arresting who have no clue that the TV version might be the U.S. one since it's more likely to get shown there than on U.K. TV.). In the U.K. they would be called "Standard Cautions" or "Reading the rights". The U.S. name derives from the SCOTUS case Miranda v. Arizona which was the ground-breaking case that made this required by all police when interrogating a suspect. Additionally, each state has their own version, which generally reads the same way (they explain your 5th and 6th Amendment rights to silence and an attorney) and may vary on asking if you choose to waive the rights upon receiving an affirmative answer that the rights were understood ("With these rights in mind, do you wish to speak to me?" is the proper phrasing). They also are read from cards (business card to index card sized) where the right is printed in English and Spanish and the suspect must sign it as part of acknowledging that their rights were read.
