Auckland has an easier climate with warmer summers (though sudden downpours of rain are not uncommon), milder winters and less wind. It also has better job opportunities, higher incomes, better access to international air-links, is more ethnically diverse. Tempering that is that housing is very expensive by world standards, inequality is greater (leading to associated social problems, particularly in South Auckland) high levels of traffic and poor public transport links (though these are rapidly improving) can make moving around the city a frustrating exercise.
Wellington is infamous for its wind and summers can be unsettled but on a nice day, the city can be sublime. Access to the outdoors is easier with fantastic walking, trail running and mountain biking opportunities available in the green belt and hills that surround the city. Multiple-day hikes and deer hunting can be accessed an hour's drive away. Wellington has a compact CBD that generates quite a buzz (high concentration of music venues, bars, restaurants etc) whereas economic activity in Auckland is spread out over a much bigger area which dilutes this to some extent. There are plenty of friendly neighbourhoods in Wellington and the level social problems, crime, burglary etc is lower compared to Auckland