While describing the style of someone or something, I use the phrase 味道, for example, 你这身打扮很有英國淑女的味道. While describing feelings of someone, I use the phrase 滋味, for example, 別是一番滋味在心頭. But in most situations, they are the same.

味道is more concrete,but 滋味 is more abstract. Ex: 想念一个人,这种滋味不好受. Missing somebody is like suffering. 今天的午饭味道不错 Today's lunch is good.

味道 is usually used to describe the taste of food.滋味 is used to the feelings of heart

味道 = taste 滋味 = flavour

滋味 can be used to describe the taste of the food,too.