I think it's all in the freshness. My parents grow their own fruit and vegetables which taste amazing. The ones you buy in the shop, having probably been left in cold storage for who knows how long, are dull and uninspiring in comparison.

I speculate that the varieties are different too. A carrot grown for a supermarket has to have properties that its home-grown cousin needn't, for instance a uniform size and shape to facilitate packaging, and not shrivelling after a week in storage. These probably win over taste when it comes to selecting the variety.