Why doesn't a "clean" car look like new?
Much of the degredation you see in the bodywork's finish is actually the result of very fine scratches and other damage to the clear coat (also known as laquer) that most cars have over the paintwork. These are visible through the light refracting off the edges of the scratch. Darker colours such as black make these sorts of marks especially visible!
What causes these marks?
A combination of factors really - general wear and tear, poor washing technique, erosion from bird droppings and many mechanical car washes are all contributors.
Can they be fixed?
Assuming the scratches don't actually penetrate all the way through the clear coat they can generally be removed or reduced through machine polishing using a hand held polisher such as a Meguiars G220 and an appropriate polishing compound. The principle is that by grinding away the clear coat until it is smooth you then get a uniform passage of light through resulting in a better looking finish. This process is often referred to as "machine polishing" or "paint correction"
The whole process (often called "detailing") is a multi-step procedure that is focussed on getting the car as clean, as damage-free and as protected as possible and the results can be very impressive since the scratches and marks take a while to build up enough to be noticeable (especially if you use high quality sealants and waxes to protect the clear coat) you can enjoy the benefits for quite some time.
If you want to see the sort of improvements you can achieve have a read of this thread on Detailing World about one of my cars that a friend and I worked on:
What's the catch?
It's worth noting however that there are some caveats to this -
The whole process including all the ancillary preparation, cleaning and protection tasks is very time consuming and labour intensive - depending on condition a mid-size saloon car can easily take a whole day or more to do, and you can't really do it piecemeal. For an example the detail in the above link took two of us two days to do, machine polishing the bonnet alone took 2.5 hours! If the hour or so it takes to do a normal wash of the car feels like a lot to you then this is not for you!
It's not cheap - you can expect to spend a few hundred pounds up front in tools and equipment and you can easily go through £40+ a time in supplies.
You can't keep repeating this process ad infinitum however since you are making the clear coat marginally thinner each time. Which brings me on to the most important thing you need to know about this process:
It is possible to seriously damage the car's paintwork if you are too aggressive with pressure, cutting compound or if the bodywork is not free of dust and grit
With that in mind it needs to be approached with caution - read guides, watch youtube videos, read some more guides and only then proceed carefully
That sounds like too much work - what else can I do?
Take your car to a professional detailler - it will likely cost you ~£100-300 but you'll likely get a better result then doing it yourself and as I said earlier if the finish is properly looked after the benefits can last for years.