You've already noted that they have similar training periods. However one thing you've omitted is the fact the British Army is not amphibious infantry, whereas the US Marine Corps is. The British Army infantry regiments consist of the Rifles (5 battalions, also the largest regiment of the British Army), the Footguards (senior infantry regiments) - which includes the Grenadiers, the Coldstreams, the Scots Guards, the Irish Guards and the Welsh Guards. The Parachute Regiment is another regiment, considered the elite in the British Army. None of the units mentioned so far are dedicated amphibious infantry. They are line infantry or 'heavy' line infantry, albeit they train as amphibious to some degree but are not dedicated like the USMC is.
The Marine Corps 'provides power projection from the sea', and while it is separate from the Navy it is more comparable to Her Majesty's Royal Marines, who have a similar role.
The Royal Marines Commando are the Marine Corps and amphibious infantry of the UK, however they are considered an elite unit among the Navy. Royal Marines Commando are trained to a far higher standard than the US Marines, with a 6 Month basic training regime and Herculean fitness standards. The Royal Marines' style of warfare is entirely unconventional with little emphasise on the use of combined arms tactics - they are trained to fight with no support, in other words. Although it's not always the case, USMC is usually an entirely conventional fighting force.
The Royal Marines attrition rate is above %80, meaning only 2 out of every 10 applicants pass the grueling training regime. They are far superior to the USMC in my opinion.