The Old World as a whole had a disease climate that was worse than that of the New World. Most famous epidemic diseases, like smallpox, bubonic plague, measles, typhoid, malaria, and yellow fever (as well as many more) have an Old World origin.

Within the Old World, diseases moved around a lot. Whenever people trade with each other, go to war or move from place to place for any other reason, diseases can spread. Consider, for example, the Plague of Justinian that spread from Asia into the Mediterranean world in the 6th century CE, or the Black Death in the 14th century which did the same and then spread into northern Europe as well.

Sometimes historians can track a disease pretty well, especially if it has unmistakable symptoms, like the buboes of bubonic plague. But a lot of diseases are trickier. When you read sources from the early modern period, for example, you find innumerable references to vague symptoms like "fever" or "fluxes," not to mention confusing symptoms like "planet-struck." But fever could be practically anything -- scarlet fever? smallpox? measles? As you can imagine, it can make it hard to see if a disease is new or not.

Also, diseases can change over time or even disappear, like the mysterious "English sweating sickness." It showed up in England in 1485, caused copious sweating and quick death, and disappeared in the middle of the 1500s. What was it? Where did it come from? Nobody knows.

To take one example of how difficult it can be to track disease in the past: we know that a milder form of malaria (benign tertian malaria) was present in England no later than the 1500s. But some historians think it was brought to England by the Romans many centuries before, while others suggest it had only just arrived from Holland.

Trade and war allowed disease to spread freely through the Old World, but there were sometimes limits to its spread. Yellow fever, for example, originated in Africa, but its spread is limited to climates where the mosquito that carries it can live. (It was brought to the Americas with the slave trade.)

On the topic of disease in the New World . . . some diseases were common to the New World and the Old before contact. Tuberculosis is probably the most well-known example, but rabies probably existed (at least in bats) in the Americas as well as in the Old World. Anywhere you have cities in the premodern world (whether London or Teotihuacan) you can expect to find diseases related to sanitation. Staphylococcal and streptococcal diseases have probably afflicted humans around the world forever.

As far as diseases with a New World origin: syphilis probably originated in the New World, although this is the subject of a truly enormous debate. (It showed up as a very dramatic new disease, called "the Great Pox" in Italy in 1495, not too long after Columbus and his men returned from America.) DNA studies seem to point to a New World origin, but on the other hand, I just recently saw an article published in 2015 that claimed to have found evidence of possible pre-Columbian syphilis in Austrian remains.

Two more diseases with New World origins are Chagas' disease and leishmaniasis, both of which are parasitic infections. Both are spread by the bite of insects, and both can cause quite serious damage. Chagas' disease remains a problem mostly (but not exclusively) in Latin America. Leishmaniasis crossed the ocean more effectively, becoming endemic in places ranging from Africa to India, where it is still a problem today.

A few very readable books:

Alfred Crosby, The Columbian Exchange (1972). This book is old now, but it has had a huge impact on environmental history, the history of the Atlantic world, and the history of medicine and disease.

Suzanne Alchon, A Pest in the Land: New World Epidemics in a Global Perspective (2003)

Mary Lindemann, Medicine and Society in Early Modern Europe (1999)

Robert Sallares, Malaria and Rome: A History of Malaria in Ancient Italy (2002)