It is not a question of which name but Brambles and Blackberries should be thought of together as one is the fruit of the other.Looking carefully at this picture you can see young unopened buds at the top center with some flowers that have been pollinated and begun to show nascent green fruitlets which will turn into blackberries when they swell and ripen.

As with many fruit there is much in the breeding and plant selection and I recommend checking the availability of various selections 

Wild Brambles

• Hedgerows and uncultivated land can become home to robust plants. They are often treated as weeds as for gardeners they have invasive tendencies.
• Long stems or branches often overflow on to paths and the thorns or spines can catch the unwary.
• Plants growing in sunshine can provide a large crop of fruit
• Bramble jelly used to be made from wild collected blackberries.
• In my opinion they make a good flavoursome crumble or mix well with apples in a fruit tart.
• Flowers attract a range of bees and wasps which is essential for pollination.