A blend is easy to explain, it is in theory rather easy, as it means one is literally blending several rums together. This is in practice somewhat harder as one needs to know exactly what they are blending to get certain results and the power of the components used.
Whereas a Solera style rum is more concerned with the aging process itself. The goal however is somewhat similar to blending; to get a consistent quality product. The process of "Solera'' which means "on the ground":
The casks with the oldest rums are on the bottom of the stack, once some is taken from these casks it gets filled up again with rum from the casks laying on top (these casks are called criaderas or nurseries). This process repeats itself over and over and the rum from the Soleras are bottled and sold. So, one is never truly sure of the actual ages of the rum. Whereas with blending you know you are combining a 5YO and an 8YO and how much of each.
People are very divided with respect to the use of Solera as they claim it to be either fraudulent or the best practice, though in my opinion adding sugar is the true culprit, though luckily rum is still relatively a free spirit so there is a way for most of us to find solace!