In general, ICs tend to be better at sinking current than sourcing current. Hopefully someone with more IC design experience will be able to answer why.
More current sinking capability doesn't automatically translate to higher heat dissipation; it's all about how much resistance the output drivers have. Mosfet drivers often have on-state resistances in the tenths of ohms and can switch hundreds of Amps without getting warm if driven correctly, but generally speaking LED drivers aren't quite this robust.
The sink or source current capability of a driver gives you an idea of what it can be used for. 60mA will give you plenty of capability to light regular old LEDs, but probably won't work very well to light a bunch of them, or to activate relays or incandescent light bulbs. It's important to also keep in mind what the voltage capability of the driver is (60mA at what voltage?) as well as to know what the total current capacity (or power dissipation capacity) of the port or chip is. The classical example given is the PIC processor; each pin can sink or source 20mA of current, but the entire 8-bit port can only sink or source 100mA of current, if I remember correctly. These numbers help you design your circuit so that you don't exceed the capacity of the device.