The best explanation is by example. Try googling for all cars advertised on the web with engines smaller than 2.0 litres that run unleaded, and have an mp3 connection and can been seen in a showroom conveniently accessible by public transport from my house.
Google just won't be able to help you with that query, not really. You have to make several searches and correlate the results yourself. On the Semantic web, you'd be able to express an interest in products for sale that are cars, and add the constraints. Every result would be useful. One or more UIs might enable you to do that, some may be specialized, others entirely generic.
An other example, creating a chart of things not normally stored in one place, say the popularity of diet coke, or country walks in a population versus the levels of clinical obesity in the same population. For these you may not use a web browser at all, but might use something more like Excel - but the semantic web gives you tools (SPARQL, RDF) for finding and manipulating the data that is out there and is accessible via HTTP.
So the point made by Bravax is not entirely true, not a lot may change - you may merely get some more useful and better mashup web sites. Or you may find yourself doing a whole lot of stuff you never thought of as being related to the web before today.
The current web has lots of alternatives for doing the same thing, say Animated GIFs, Flash, Silverlight, DHTML etc. For putting data on the semantic web there will be a range of tools and formats. RDFa is a good one, a more general type of microformat, but you could provide a dump of the whole database, expose a SPARQL endpoint, use a microformat or a proprietary HTML structure and add a transformation, there will be many tools to suit different cases.
So Vartec is also partially right, you may use RDFa and eRDF but you could also use a whole lot of other things for publishing data.
Note that there is a lot over overlap between the semantic web and another simper concept called Linked Data. How they relate to each other isn't clear, but my perception of it is that the Linked Data web is what you need before Semantic Web tools and techniques have anything to do. Linked Data is about data, the semantic web is more about processing the data, reasoning over it and handling issues like trust reliability and such like. Essentially the bottom few layers of the technology stack.