The difference between these is that brand equity is just one part of customer equity.
Customer equity essentially refers to the value of customers' loyalty to a particular company. The more that customers feel connected to a certain firm, they more customer equity it has. One of the three parts of customer equity is brand equity. Brand equity refers specifically to the value that customers place on the brand itself (as opposed to the value of the product). In other words, if a customer thinks that a shoe is worth more because it is a Nike shoe, that is brand equity.
Customer equity includes brand equity and is, therefore, a broader concept.