Tactical EMS (or TEMS) training teaches the medic or tactical operator to provide basic medical care and injury/disease prevention in the tactical (i.e., SWAT, special operations, etc) environment. SWAT teams serve a variety of roles to include hostage rescue, high risk warrant service (you've got that drug dealer who is known to be armed and has assaulted officers in the past, you may want to send SWAT to pick him up rather than your garden variety detectives), and special security, such as that provided to medical teams that went to areas affected by Katrina. Some SWAT teams have medics on the team who are taught team movement and SWAT operations as a way to understand team ops and provide better care. They may or may not be allowed into a "hot zone" and may or may not be armed. A lot of these medics, particularly if they are not already law enforcement officers, may train with the team but do not go into the "stack" when making entry into a potential hot zone and may not be armed. They may wear body armor and identification but will be staged in the "warm" zone, ready to respond in as needed. Some teams have medics who are also operators, i.e., have a role in the law enforcement aspects of the operation, and therefore carry weapons and arrest suspects, etc. and will go in with the stack. Frankly, having your medic in with the stack is risking an important resource for very little benefit (my opinion, has not yet been shown in randomized prospective trials ).
Many teams do not have medics as part of the SWAT cadre. These teams may stage an ambulance some distance away, ready to respond in to the scene as needed. These medics typically are not tactically trained.
From the military standpoint, special ops medics tend to be operators and therefore armed and responsible for aspects of the mission other than just medicine. They also have a larger role in primary care and preventive medicine since the SOF teams may be far removed from definitive care, and evacuation may be impossible or may jeopardize the mission. 18D (Army SF medics) are typically trained not only in advanced trauma care, but dental care, veterinary medicine, and food/water procurement and purification as well. Medicine is an important part of some SF missions, which may involve training indiginous forces. The 18D helps to win hearts and minds by providing some basic medical care to these folks.
Military medics are trained in tactical combat casualty care (TC3), which gives them tactical awareness and medical skills particular to the battlefield for rendering care while still potentially in some amount of danger. With this in mind, the best medical care is overwhelming firepower, and putting rounds downrange is often the most important thing that a medic can do in a hostile fire situation. 91W (combat medics) are trained in this.
Tactical medic training varies by institution. Where I did mine (Blackwater), tactical operations, building clearing, weapon usage were large parts of the training, but then again a lot of us were very experienced medical providers. Tactical medicine is a combination of several aspects of care. TEMS medics are first and foremost first responders, and trauma care is the cornerstone of care in this environment (traumatic injuries, as you might expect, are common). There is an emphasis on care in austere environments with low light, little space, and little equipment. As the first medics in the door, they must be firmly familiar with triage for care and triage for evac (2 different concepts, really). Team wellness and preventive medicine are important, and in the civilian realm this equates to proper rehabilitation of operators on a long standoff and ensuring that everyone is rehydrating properly. The TEMS medic may have to deal with minor illnesses in operators while on a standoff, such as minor cuts and scrapes, headaches, dehydration, etc. The TEMS medic serves in an advisory capacity as well, recommending equipment and policy that may reduce injury among operators. For example, the medic may recommend that operators carry their own "blowout bag" that contains some dressings and other supplies so operators can treat themselves or a fellow operator when the casualty can't be immediately evacuated (due to hostile fire) or the medic can't get to them. He may also recommend rotation schedules for operators to ensure that they are rehabilitating properly, particularly in inclement weather. The medic may also train the operators on some basic self- and buddy-care. He may also make sure that everyone's tetanus shot is up to date, and should be familiar with any underlying chronic illness or injury that the operators have. The medic may be asked to get on the phone in a hostage situation to try to provide some care over the phone to hostages and suspects while negotiators are trying to end the standoff. They may be able to instruct the hostage or suspects in some bleeding control or airway maneuvers as well as basic treatment for shock. This may give the medic an idea of the number of casualties inside and their condition, enabling him to request civilian resources as needed. (I should emphasize that medics are NOT routinely negotiators.) Medics also serve as an on-site consultant to the SWAT team, recommending for or against things like tear gas or distraction devices (flash-bangs) in particular situations if there may be medical concerns with their use.
Situational awareness and self defense are taught from a tactical perspective, and usually involves weapons. There may be some items to glean for civilian EMS use, but this isn't really a "self defense for the street medic" type course.
Hope this sheds some light.