ASX vs NASDAQ

I've never used hot copper before so please let me know if there is somewhere else I should ask this.

Anyway, I know this is a dumb question but is there any difference between buying shares for Immutep in ASX vs the NASDAQ?

I am from Australia but live in the US, and through my broker I can buy ASX stock and I can obviously buy NASDAQ stock too. If I was looking to invest, should I just look at the one with the lower market cap? or go with the one with the lowest fees? (NASDAQ due to foreign transaction fee with ASX).

Was just looking for someone to enlighten me as to if there was one exchange which would be better to invest in?

Thanks