Barrow originates from old English, it means to bear a load. The original barrows were implements to allow people to carry more stuff than normal and were designed to be carried by two people (imagine handles on both ends). A wheel was added later, thus wheelbarrow.

Barrel comes from French, and implies a fully-enclosed space for storing things, like wine.

They are similar things, but practically different and originating from different languages. Though, that’s never stopped modern English from mashing two disparate things together before. And there have likely been many instances where someone carried a barrel in a wheelbarrow. :)