Blackberry and bramble are the same thing. They have fruit like black raspberries, and spikey stems and leaf veins - although you can get domesticated thornless ones. They are climbing / creeping plants so they ramble over nearby supports rather than growing upright without support.
Raspberries have very upright stems / canes which grow separately from the ground and have much smaller bristles on than the bramble's vicious thorns. The leaves are softer and thorn-free. Bramble leaves are not quite so pointed (as shown in the pics).
Blackcurrants are totally different with smooth, round fruit on a traditional bush-shaped plant.
I am always confusing the names as I have both on the allotment, so I tend to stick to 'bramble'.