Notice that in that sentence "I will show you around" that you are normally either hearing the schwa sound in "around" or the person is dropping the weaker vowel altogether by saying "round". As Chris says, they can be used interchangeably so there is no harm done, but the issue is that there is a strong vowel sound 'u' in 'you' followed by the weak 'a' so the speaker will not pronounce the 'a' in 'around' normally, if at all. isn't pronunciation fun? haha.

Short answer = Yes, they are used interchangeably. If you hear people say it they mean exactly the same thing. Long answer = There is an ongoing debate amongst people who love grammar about which to use. See this http://www.dailywritingtips.com/round-vs-around/ for an example of this discussion. Sometimes people talk about the differences between British and American English, but this is one instance where globalisation means this difference is being eroded anyway.

For the purpose of translation, the proper word is "around." But in speaking, as Michael K said, because of the sound of the two words together, the "A" sound is not heard unless the speaker pauses between the two words.

Around is correct, round is colloquial.