Well, as for me показывать is to really show something physically. Like when you show a thing or when you show how to do something by doing that by yourself first and the other watches and learns. Предъявлять, well it's really used mostly in set-expressions like предъявлять обвинения, доказательства, документы etc. This word has some shade of a negative, bad meaning. It is even used as a synonym to обвинить (accuse) if it's used as an intransitive verb. For example:

Ты что мне предъявляешь? = Ты в чём меня обвиняешь? = What are you accusing me of?