Both cities belong to the oldest settlements in Australia and try to preserve old building and sights.
In Sydney, the district "The Rocks" right by the Sydney harbour is the oldest part of town which has been restored during the last decade. You find small museum as well as souvenir and other shops, nice old-fashioned cafés and restaurants as well as quite contemporary upscale shopping opportunities. You should take part in a guided walking tour through the Rocks to learn everything about the life of the early immigrants.
Melbourne's history is connected to the figure of Ned Kelly, the famous bushranger. You can observe his self-made armour and more relicts in Melbourne museum and visit the museum of Old Melbourne gaol, where Ned Kelly was imprisoned before he was hung.
Like most Australian cities, both Melbourne and Sydney are situated near the coast.
Sydney has several beaches in suburbs, for example the famous Bondi Beach, but no less beautiful (and crowded) Manly and Coogee Beaches.
Melbourne has one beach located near the city center, in the district of St. Kilda.
While Bondi Beach is famous for surfing, St. Kilda Beach is blessed with big waves, because Melbourne city is located by a bay, Port Phillip Bay. Melbournians prefer to drive a little farther out of town, up the Great Ocean Road, to Torquay (Bell's Beach), Lorne or Apollo Bay.
You should find everything you're looking for in both cities. But if you're looking for shopping malls rather than prowling the city centre, you will probably like Sydney better. The Queen Victoria Building (QVB) looks old-fashioned but offers a great variety of modern shops and cafés, besides its impressive Victorian architecture. Another shopping centre has been opened in Darling Harbour, Sydney's up-and-coming trendy district.
Melbourne's Central station has a big shopping centre with dining and entertainment options ― a cinema occupies the top floors ― but it lacks the atmosphere and architecture of the QVB.
Business versus Culture/Leisure ― this is how I would break down the atmosphere in these cities to easy terms. Sydney CBD is busy with the financial sector workers and dominated by the big office buildings, whereas the Melbourne area is filled with restaurants, cafés and art. But this is just my personal impression. While both cities have their trendy spots, party miles and entertainment venues, Melbourne offers a wider variety of multicultural cafés and restaurants from different immigrant home countries: Greece, Italy, Ireland, France and various Asian countries for example.
A big plus for both cities. You get to experience a lot of cultural events there all throughout the year, in addition to the cultural 'classics' you can always visit: In Sydney, you can of course have a look at the Opera house ― if you're not into opera performances, you can take a guided tour through the Opera house during daytime, and you'll also learn a lot of interesting things about its remarkable architecture.
Melbourne has the amazingly central and amazingly concepted Fed(eration) Square – it's THE public place to be. A lot of events happen here on a small stage or are transmitted on the big video screen, and on a normal day, there are street artists entertaining the crowd that always gathers there.
The best way to get through Sydney is actually by bus ― there is a big network of bus lines, but for someone who's new in town, in can be a little complicated. There are also a few CityRail train lines and regular ferry services departing from Circular Quay to a number of beach suburbs around the Sydney harbour.
In Melbourne, tram lines are still in use, which adds to the city's nostalgic charme. Of course, city train and bus lines exist as well. The tram system works well, and there is an extra tourist tram ― it runs on a circular route around Melbourne's CBD and also plays a recording with information on the sights in passing. The so called City Circle Tram is free; the tourist tram line can be easily recognized and distinguished from the normal, chargeable tram lines because it uses old-fashioned, dark red tramcars.
Both cities have a stadium for major events and concerts, a few musical and a few more acting theatres, cinemas, and a lot of live musicians bringing life into the pubs even on weeknights. Have a look at one of the free city magazines for a weekly overlook ― there'll surely be something you'll like. During summertime, both cities usually have big open air events, either in the Botanical Gardens or in the harbour area (of either city).
Although Sydney had the famous Olypic Games in 2000, Melbourne regularly hosts internationally renowned sporting events like the Australian Open or the Formula 1 Grand Prix, just to name a few.
Basically, it's head-to-head. Both cities offer everything you might look for in a modern and diverse metropolis, but they are still different in the details and the atmosphere. Which city you like better in the end is based on personal preference.