What are the differences in the wiring standards between the USA and the UK?
Many, many things. 

First up, voltage. For historical reasons just about every domestic installation in the UK uses the same European standard voltage of 230V for every socket. In turn, almost all sockets used indoors are the same size and shape and deliver the same maximum current. (Some farms have three-phase 415V supply, but  I don't think that's often run to the house itself...!) 

Just in itself that introduces a whole level of complexity as the higher voltage is much keener on using a human being to find its way to earth. 

Secondly, current. UK domestic power sockets are all rated at 13A, whereas in the US there's typically different sockets for different kinds of jobs. However, these sockets are joined in a "ring main" which is typically rated at 32A in total for all the sockets on the ring: so in theory connecting live and neutral on a ring main can deliver 230V * 32A = 7.3kW. 

Makes for a nice bang. So it's wise to be careful.

Power in the UK is distributed from the supply company's meter to a "consumer unit" (these days, RCDs and circuit breakers) from whence the ring mains for sockets, lights and heavy equipment are powered. Like sockets, breakers are double-pole. The power company generally only provides a single-phase supply from their meter, it's an offence to tinker with anything upstream of that.

Earth is typically a earth rod (TT), separate earth from the supplier (TN-S) or combined and separate from the supplier (TNC-S) combined with separate connections to the water main (if it's metal) and the gas main. 

Most houses have two ring mains, typically upstairs and downstairs, and for convenience most have an upstairs and a downstairs lighting radial circuit. Electric cookers typically and water heaters (immersion heaters) typically get their own circuits rated at 32/40 amp and 20 amp respectively Lighting, the ring mains, and high-current circuits use different sizes of cable and m.c.b's (miniature circuit breaker).

The UK has been described as one of the most heavily fused countries on earth. As the ring main can in theory deliver 32A to the socket, each appliance has its own fuse built into the plug. That, along with the higher voltage and current, is one reason why the UK has such big plugs. These fuses are (these days) clipped into their own holder which can be (reasonably) easily extracted without taking the plug apart; they come in 3A, 5A and 13A sizes.

There's more, much more: bathrooms and other wet areas have their own special restrictions known as a special location, there's regulations about how wires can be joined together, how many sockets can be on a ring, the distance between sockets, the number of spurs permitted, and so on and so on. A three year apprenticeship is a good start, as is careful digestion of the Wiring Regulations BS 7671

What can be done by a DIYer? Notionally buildings regulations have always required a "Competent Person" to do any electrical work. Since the introduction of Part P of building regulations a recognised contractor needs to inspect major work or work in bathrooms and kitchens.