First, let's get our terminology straight. A multi-core design is a single chip with multiple CPUs inside it, with at least some memory shared between the cores:

Multi-core parts span a very broad range, from the larger, 64-bit parts that have multiple identical CPU cores to the ARM big.LITTLE architecture, which incorporates both high-bandwidth CPUs and power-conscious MCUs in the same package. Recently, multi-core MCUs have also become more commonly available. Graphics processing units (GPUs) can also be grouped into the multi-core category.

A multi-processor system is one where there are multiple processor chips in the design. For the purposes of our discussions here, these chips can reside on the same printed circuit board assembly (PCBA) or different PCBAs distributed throughout a larger system:

Both multi-core and multi-processor topologies can be found in many different places, such as smartphones, small networked sensing devices, industrial automation equipment, test equipment, medical equipment, appliances, and of course, a range of computing devices, such as desktops, laptops, and so on. 

There are many different driving forces for using these two different topologies, beyond a simple need for more or faster processing. Sometimes, a system needs to come online instantly, without waiting for a full general-purpose operating system (GPOS) to boot. Occasionally, there are regulatory requirements that are easier to meet by segregating system functions into multiple cores (and code bases) so that only a portion of the total code (or system) is required to go through a stringent review. There could be electro-mechanical considerations in a system (such as long wire runs to motors/actuators or sensitive analog signals) that are best addressed by having a processor in close physical proximity. In high-reliability systems, redundancy is very common.

Now we have a general idea of the terminology, let's get into some additional details and use cases for these systems, starting with multi-core designs.