Properties of berries that are not shared by strawberries
Botanically a berry is something quite different from what is commonly termed a berry in English.

From that Wikipedia page:


  a berry is a fleshy fruit without a stone produced from a single flower containing one ovary


...


  Berries so defined include grapes, currants, and tomatoes, as well as cucumbers, eggplants (aubergines) and bananas


On the other hand, many things called "berries" (including strawberries) are actually aggregate fruits, which are


  a fruit that develops from the merger of several ovaries that were separate in a single flower


Therefore, the botanical difference is the originating tissue of the fruit: a single flower/one ovary, versus a collection of ovaries grouped together.

These botanical groupings don't have much relevance to practical use (i.e., cooking/nutrition) and there aren't other properties that are definitive for what a berry is and is not. For that reason, there is nothing wrong with referring to strawberries, raspberries, etc, as "berries" in cooking: berry simply has different meaning in botany and culinary fields, just like the many culinary "vegetables" that are botanically "fruits."