What's the difference between a steak cooked rare and a steak seared rare?

You can cook a steak to rare by using any method of cooking the steak, grilling, broiling, pan frying over medium high heat, roasting probably even by simmering (pan frying over low heat) or boiling. In most methods of cooking like grilling, broiling and pan frying the outside of the steak does get seared which helps to retain the juices in the steak. Simmering, boiling or baking a steak will still get to medium rare, but the juices may leach out giving you a steak which may be dryer than when a steak is seared. Searing the steak also gives it a better flavor while baking or simmering give the steak gives it a sort of boiled flavor. Any thickness of steak can be seared and cooked on a grill, but sans a grill, thick steaks are best seared in a hot skillet then finished in the oven. NOTE: Once the steak is cooked to your liking, remove to a warm plate, tent with foil and allow to rest 3-7 minutes (depending on thickness) before cutting into your steak. Resting allows the juices in the steak to redistribute which will give you a more evenly cooked and juicier steak. While resting, the internal temperature of the steak will rise by about 5 degrees. Resting works well for chops, steaks, roasts using all types of meat. I have even found it works with baked and broiled fish.

Steaks (even rare ones) are cooked in a way so that their internal temperature reaches high enough to kill any bacteria inside of it. Searing is an extra process added where the outside of the steak is cooked at a very high temperature in a short amount of time to produce some charring. Searing is used for both appearance and taste (some people like the taste of char).

There really shouldn't be any difference. Searing indicates sealing the surface of the steak over a very high heat, often on a griddle plate or griddle pan and honestly this is the way that all steak fit to be eaten rare, should be cooked.

Bacteria inside beef is very rarely a concern. It is the bacteria on the surface of the meat that is dangerous. On a steak it is killed instantly as the meat hits the hot grill/pan surface. With burgers the infection is minced throughout the meat, which must then be cooked thoroughly to be safe. Truly, red or pink burger meat can make you sick.