My understanding of it:
In the US there are 2 types of visual approach, one is called the "visual approach" and needs a 1000' ceiling and 3sm vis at the airport - the other is called a "contact approach" and only needs 1sm vis and an airport which must already have an IFR approach procedure. ATC may offer a "visual approach" (indeed, it is up to the pilot to say so early if it isn't acceptable), but the "contact approach" may only be initiated by the pilot. Both procedures are IFR and do not affect the IFR flight plan.
In the UK you may fly a visual approach (ie. leave the IAP profile) if you can maintain visual contact with the surface - you may descend below circling minima if you can see the approach lights and/or the runway (the published circling minima and RVR/Vis apply, but this may be operator specific?). And, as in the US, clearance for a visual approach does not affect your IFR plan.
That's the way I read it, anyway: usual health warnings apply (!).............