The law aside, one of the prime aims of the archaeologist is the preservation of artefacts. If it's thought that artefacts will be at a higher risk of deterioration in the ground than if they're removed, then they'll be removed.
The opposite is also true. If they're thought to be safer where they are, then they'll be left until science has found a suitable alternative method of excavation/preservation. Sometimes this never happens.
Whatever happens to them, they are carefully plotted, examined and recorded so that future generations can learn from their discovery. That is archaeology.
Grave robbing is done by the indiscrimate 'digger' who sees the goods and takes them for his/her own gratification (financial or otherwise - look what I found!). He/she rarely records accurately, if at all, and anything we may learn from the context of the 'grave' (such as whether the jewel was found near the throat or the waist, for instance) has been lost.
Oh, and anything you do find, treasure trove or otherwise, has to be offered to the landowner (from whom you should have sought permission to dig in the first place), because it's technically their property.