Height mostly. There's not actually that much difference - only a foot between UK rolling stock and a TGV Duplex but if you're going to collide with a bridge, the result is the same whether it's a foot or a yard. Ceilings on TGV Duplexes are pretty tight so knock six inches off of each floor and I'll just be glad not to be 6 feet tall. They did try double decked trains in the UK but it was absurdly cramped.
UK railways were the first in the world and were built to fit the small steam locomotives of the day. Therefore Network Rail limits the height of rolling stock to 13'1". European loading gauges were agreed at a minimum of 14'1" in 1913 and France had to do a lot of work to improve its infrastructure to match the standards set by the Kaiserreich, but labour was cheap then. The UK wasn't party to the Berne Convention - it had no need to be as apart from a handful of vehicles that used the ferries, nothing from the continent would ever enter the UK until the Channel Tunnel was built in 1994.
Could we improve the loading gauge of our existing lines? In short it would cost a fortune. In order to install electric wires on the Great Western Main Line, Network Rail has spent a considerable sum of money raising bridges and lowering tunnels in order to fit electric wires in and maintain the 13'1" clearance (bear in mind that you have to keep the wires a safe distance from both structures and the roof of the train otherwise it'll arc). The UK is a very crowded country so you've got more bridges to do.
Any engineering work done has to fit around existing traffic. Everyone hates replacement buses, imagine having the things every weekend until further notice. In 1998 Railtrack started the West Coast Main Line Route Modernisation. After many overruns the project was finally finished (after being descoped) in 2009. The cost had gone from £2.5bn to £14.5bn in four years (2002 prices), driving Railtrack into liquidation as it went. That was for a project that yielded a fraction of the capacity that HS2 offers.
Adding extra lines to the existing route won't work either. The WCML wiggles it's way through the country, having to avoid the country estates of influential 1837 landowners. Development has been built right up to the route, so you'd still have to demolish loads of houses. You'd still have to rebuild Euston and demolish the same amount of Camden to accommodate it.
Just to illustrate how much capacity the project gains, there will be room for 13,300 commuters and 21,600 long-distance travellers out of Euston per hour. That compares with 5,500 commuters and 5,800 long-distance travellers now. It also releases some capacity out of St Pancras and King's Cross. You'd need to build two motorways to shift that number of people (just think what sort of land take that would be).