Two fundamentally different ways of delivering heat
A heat mat is a flat pad fixed under or against the enclosure that warms whatever it touches. Heat travels into the floor above it and into the belly of an animal lying on that floor, but the surrounding air barely warms at all. An overhead lamp works the opposite way: it radiates heat downward from above, warming the air, the surfaces in its beam, and the animal basking beneath it, much as sunlight does. That single difference, contact heat from below versus radiant heat from above, explains almost every practical distinction between the two, from which species each suits to how the temperature gradient in the enclosure behaves.
Which animals each source suits
Ground-dwelling, mostly nocturnal species are the mat's home ground. Leopard geckos, many snakes, and other animals that shelter in warm burrows or under sun-baked ground take in heat through their bellies and can do well with a warm floor zone covering roughly a third of the enclosure, leaving the rest cool. Diurnal lizards are lamp animals. Bearded dragons, chameleons, monitors, and day-active skinks are built to bask, raising their bodies into a beam of heat and light, and a mat cannot give them the raised basking surface or the bright, warm focal point their behaviour expects. Arboreal species that rarely touch the floor gain almost nothing from a mat at all. Matching the source to how the animal heats itself in the wild settles most of the debate before it starts.
Gradients, air temperature, and the night
A lamp builds a three-dimensional gradient: hottest at the basking surface, cooler across the floor, coolest in the shaded end, which lets the animal fine-tune its temperature by moving inches. It also lifts the overall air temperature, which matters for tropical species that need warm air, not just a warm patch. A mat leaves the air at room temperature, so in a cool room the animal may have a hot floor and chilly air, a combination that suits some burrowers but leaves basking species unable to raise their whole body temperature properly. At night, a glowing lamp disturbs the dark cycle; keepers who need night heat from above use a ceramic emitter that gives heat without light, while a mat can run quietly around the clock under thermostat control.
The safety points that apply to both
Both sources can burn an animal when run uncontrolled, and both must run through a thermostat, no exceptions. Mats have a particular trap: an animal lying on an unregulated mat cannot escape heat rising through its belly as effectively as it can walk out of a lamp's beam, and mats under heavy enclosures can develop hot spots. Lamps have their own traps: a bulb an animal can climb against will burn it, so fittings need guards and sensible distances, and the basking surface temperature must be checked with an infrared gun rather than guessed. Whichever source you choose, measure what the animal actually experiences, at belly level for a mat and on the basking surface for a lamp.
Choosing, and combining, for your setup
Start from the species, then the enclosure. A basking lizard in a tall vivarium points clearly to an overhead lamp as the primary source. A leopard gecko or a burrowing snake in a low enclosure points to a mat under one end. Some keepers combine both, a lamp providing the daytime basking spot and gradient, a mat holding a gentle floor warmth overnight, each on its own thermostat. For another view on a specific combination, put a follow-up question to KinsLe's Kimmy; she answers using the KinsLe Answers library.