in some occasion,both are same:cheap.i guess you already know.but when they are used in evaluation ,便宜 can be applied in food,and some other goods ,however,廉价 dose not often emerge in describe food.(这家餐厅很便宜，他们的菜很便宜:the restaurant is cheap,and their dishes are cheap.) 廉价 low price：廉价商品（goods)，这件衣服廉价出售(this cloths is selling at low price)well,you can interpret that 便宜 is more likely something have low value,then sell cheap,廉价 on the other hand means something is worthy ,but have no other choice to lower price ...the single word 廉 sometimes gose with 洁（廉洁），on their opposite,it is corruption.so 廉洁means clean hands (为官清正廉洁：to be a clean hands and honest government officer.) i want to talk more about 便宜(不要贪小便宜 ,否则你会因小失大）do not take little advantage，otherwise you will lose more ) ....at last 便宜can be pronunced( bian yi) mean easy and convenient.

I'm a Chinese student.In my opinion, 便宜is more used in spoken Chinese and 廉价 is more for writing

最简单的区分： 便宜，是相对而言的，一个商品通常或别人或原价是１００元，现在为80元，因此，现在便宜。也就是说，价钱便宜，并不意味着商品不好，虽然绝大多数情况下，便宜没好货。 而廉价，就是指的那个东西不值钱，你想卖高价也没用，因为价格高了不会有人买。 便宜一定是相对其它而言的，比如苹果，人家卖十块，他卖九块，他的便宜，并不能说他的廉价。廉价除了指价钱低之外，更多的是指物品不值钱。

”廉价“ and "便宜” both means cheap. each of them could be used to discribe the material objects which are cheap. only "廉价“ could be used to discribe a human being who is cheap. eg.1 the furniture is quite "廉价”/“便宜”. eg.2 She is very "廉价”. as to "廉“，we usually don't use it alone.

There's no big difference between 便宜 and 廉价,you can use one to replace another one.廉 is like the character 'a' in the word 'about',we often use it with '价' together.They both means cheap.