As a word, the negative connotations come from an ignorance nowdays, rather than it becoming a racist term itself.
Brit and Aussie, if you call someone a Brit or an Aussie, it's probably because they're British or Australian. It's rare that you get something like that wrong. They are very distinctive countries in terms of how they sound.
Paki is different in that it reffers to everyone in south(east?) Asia, OR even worse, people with a complexion even similar to that. Yes, that DOES happen. It happened to me once. I got called a Paki in my childhood, even though I am of mixed (British/Jamaican) descent.
As a whole, 'Paki' is used in a negative context more than 'Brit' and 'Aussie' are.