I don't keep much food in the house, and generally use the fridge as a general purpose store for all kinds of food, as a way to slow down chemical reactions and lifeform growth alike, as well as spending less time considering the type of storage for a particular item. However I've recently experienced some potential pitfalls, some honey has solidified into what could be a crystalline sugar solution, and a recent, official peanutButter/marmite mix developed a thin, non-fury, jelly like, yellow layer on some of its surface, apparently these are due to keeping the foods in an environment that is too cold. Is it true that there are trade-offs either way regarding heat? 
Many food items, particularly those containing oils, but many others too, will change characteristics when cold. The good news is that keeping honey and most other things at fridge temperature does not generally affect flavour, but it may affect texture permanently. Oils and other substances that can go rancid will generally keep better in the fridge than at room temperature as the lower temperature will help slow the chemical reactions that make oils rancid. You have observed a characteristic change with your honey, where it has gone crystalline because you have lowered the temperature to a point where the sugar solution is now saturated and crystals form. The crystals should re-dissolve when heated gently to above room temperature. Honey is one substance that will usually store more or less indefinitely at room temperature without spoilage. Peanut butter is another one that you might see, where the oil may separate and solidify at fridge temperature. The suspension of ground peanuts in oil may also become lumpy and/or difficult to manipulate because of the solidification of the oils. In some cases (particularly things with a high oil content, like Nutella) the solidification will result in permanent texture change as some of the components will separate from the oils, clump together, and can not be easily returned to a homogeneous mixture by your regular stirring methods. In some cases it may be difficult to tell a state change from a contamination. For instance, oils generally go from clear to translucent, and may form globs/lumps that might be mistaken for bacterial colonies. Separation of liquids from gel-like substances may be a result of storage at too low a temperature or as a result of bacterial or fungal contamination. However, if you have any doubts about the safety of a particular food it is always best to throw it out 