They are the same.

What is the difference between 鞋and 鞋子?

Well,we can say  旅游鞋

                                 板鞋

                                 拖鞋

             but no        旅游鞋子

                                 板鞋子

                                 拖鞋子

In conversation we can say "我的鞋呢？“

                                                    “我的鞋子呢”  They are the same..

What is the difference between 鞋and 鞋子?

Here, 子 is used as a dimiminutive, like English's kitchen --> kitchenette. In your example, it also conveys a compliment.  Until the end of 1912, small feet (for women) is considered a beauty (hence the practice of foot binding).  It can also be as a sort of endearment.

In other cases, other meanings of 子 applies.   天子 = Son of Heaven, i.e. the emperor although one can argue he is "little heaven". With terms 老子, 孔子 etc , it is an honorific term for a person.