A very generic question, but I'll give my two cents.
There are varying construction methods used to build skyscrapers. The selection of which depends on a variety of factors including location, climate, the type of ground, the height of the building, the use of the building, etc.
You'll also notice regional and national preferences in construction styles and methods.
The concept of a 'core' only arose later in high rise construction. The core is typically a central component of the tower which houses mechanical features, elevators, stairs, plumbing and electrical utilities.
For a long time, steel construction consisted of what is essentially many boxes stacked on top of each other. With this type of design, the load is evenly distributed across all of the columns, and there is no technical need for a central core beyond practicality and space planning. For example, 14 Wall Street features an elevator bank and utility "core" that is actually located at the perimeter of the tower, and if you look at photos, the western face has blocked out windows.
Note the opaque windows on the tower, denoting where the elevator shaft is aligned.
With concrete construction, there is usually a similar lack of central 'core'. Again, for practicality sake, you'll usually want to put your elevator shafts in the center of the building, but that doesn't make it a true 'core'.
For example, most recent residential high rises in NYC have been built out of concrete. In these, instead of putting all of the plumbing and electrical in a central core, and then branching out to each floor, they will instead align the bathrooms and kitchens on each floor, and then run the utilities in multiple voids eliminating any horizontal pipes.
The World Trade Center brought a new concept with steel construction with it's central core and perimeter columns providing a column-free interior space. Here all plumbing and utilities as well as elevator banks ran through the un-reinforced core. Post 9/11, we learned the drawbacks to this construction method.
With newer construction, such as the new WTC complex, a hybrid concrete / steel approach is used. In these buildings, you'll see there exists a highly reinforced concrete core, surrounded by perimeter steel and columns. This allows for the same column free interior floor space that the original WTC towers offered, while mitigating the risks of collapse. It's arguably the safest, but most expensive way to build.
Ultimately though, there is no quick easy answer as to why one building has a core while another does not. Typically though, you'll find that residential buildings feature smaller floorplates, with lower ceilings, and lots of internal walls, which allow for cheaper concrete construction with no core. Meanwhile commercial development requires larger floorplates with taller ceilings and a demand for no interior columns interrupting the open undivided office space.
Also, office buildings will often fit the bathrooms into the space between elevator shafts on floors where the bank does not stop (using segmented elevator banks for different floors, a technique pioneered by the World Trade Center), a practical use of this space, which further bolsters the 'Core' design by housing all plumbing and utilities within.