Going just by definition, I would differentiate the two as follows.
Brand Awareness: How much do customers know about your brand. Have they heard about it, do they think about it, do they talk about it?
Brand Salience: Do customers think your brand is the ‘gold standard’ in the category. Is it the first brand that comes to their mind when you talk about the category?
A brand can have a higher level of awareness from time to time owing to the promotional campaigns they do or news about the product, however Salience is achieved through consistent delivery of value. For example, Samsung Galaxy 7 is right now pretty high on brand awareness scale due to reports about spontaneous combustion of the phone. However it doesn’t mean people will go and buy it. They will still go and buy iPhones. 🙂