A 'show-off' is someone who performs or does something, often very well, but with the intention merely of showing that he/she can do it. They may also do it in a particularly flashy or over-demonstrative way.
'Show up' has two meanings. The one which is relevant here is partly related to showing off. If someone attempted to show off and then someone else came along and did the same thing better, but in a less showy way then the latter person could be said to have 'shown up' the other person. To put it another way, it means to reveal someone as being a bit of a fraud or a fake.
The other meaning for 'show up' is just to arrive at something like an event or an appointment. For example, 'he showed up ten minutes late.'