in laymans terms, a road bike is faster, designed for going fast for a long time. So geometry is geared towards being in the drops, weight is kept to a minimum and gears are big (40 odd tooth big ring etc). CX bikes are designed for going round muddy grass circuits for about an hour as fast as possible, gearing is smaller (about 36 big ring) Geometry is a bit more mountain bike like (lower top tube etc).
A dedicated "real" CX bike will not have braze ons for water bottles, mud guards, panniers etc, though with a £1k limit most bikes will accommodate this (or use clips).
I use a CX bike (kona jake the snake) for commuting every day and occasional racing as well as big road rides/racing. To me, its the perfect road bike for a mountain biker, I can go off road if I fancy it, i can carry big loads on the pannier, its got a granny ring so I can spin up hills and brakes that make sense to me (ish, its canti's).
road tyres on a CX bike makes it basically the same as a road bike as much as the amateur rider will ever need IMO. I can tell the difference when I ride a "proper" road bike, but I think thats the gearing as much as anything? I'd go for a CX bike for the job required, you could always change the rings if you feel under geared?