I got this information from
a lecture on Acute Coronary Syndrome by Professor Edward P Sloan, a Chicago based emergency physician.
and also from
Short answer: thrombolysis is less effective due to the difference in clot types (white vs red) to the point where the risk/benefit is unfavourable.
"thrombolytic agents have not been shown to provide benefit in patients with UA/NSTEMI; in fact, there is a trend toward worse outcomes"
-STEPHEN D. WIVIOTT, M.D., Instructor of medicine at Harvard Medical School - investigator with the Thrombolysis in Myocardial Infarction (TIMI) Study Group.
STEMI's are transmural infarction from full occlusion of a main vessel.
NSTEMI'S: not transmural infarction and caused by partial occlusion of a main vessel.
The composition of the plaques are different:
STEMI plaques are fibrin and red blood cell rich, hence the term red thrombus.
NSTEMI plaques are mainly platelets
STEMI's are 100% fibrin-rich, hence the role of fibrinolysis.
NSTEMI's appear about 29% fibrin, but 71% platelets - hence the crucial importance of antiplatelet drugs in nstemi's.
RJ frink on why there may be a trend towards negative outcomes of thrombolysing NSTEMI's:
Many of the multiple, chronic Ulcerative Plaques (UP's) found in patients with Acute Coronary Disease contain intraintimal and mural thrombi, with plaque contents incorporated into the body of the thrombus. An intraintimal thrombus acts to seal the UP  and to prevent the discharge of plaque contents and plaque toxins. A thrombolytic drug, by causing lysis of intraintimal thrombus, reverses the normal repair responses (Chapter 9), and reopens what is essentially a sealed UP [148,260]. Multiple UPs may be present in all 3 major coronary arteries , and may be reopened by these drugs and discharge plaque toxins to the entire myocardium. This could result in global myocardial dysfunction in spite of adequate coronary blood flow [222,261]. In addition to reopening the core, the lysis of thrombus also releases any plaque contents and toxins contained within the body of the thrombus, adding to the toxins liberated from the reopened core itself. Microembolic obstruction of the microcirculation could also contribute to toxic injury by obstructing antegrade, collateral, and anastomotic flow, as discussed above .
The lysis of intraintimal thrombus also allows blood to re-enter the now empty, or partially empty, necrotic core . This reentry may change the dynamics within the core. For example, the reentry of blood could lead to dissection of blood along cleavage planes (Figure 7, Chapter 4), the formation of blind pockets (Figure 20), or the disruption and/or dislodging of the fibrous cap to form an occluding flap of tissue (Figure 20). Administering a thrombolytic drug is also associated with the activation of platelets [263,264]. Activation of platelets may lead to the formation of a new intraintimal thrombus in a newly reopened necrotic core, and lead to an occlusive thrombus and acute events. Therefore, thrombolytic drugs may convert a relatively stable UP into an unstable UP, resulting in fatal and non-fatal AMI. The mechanisms described here could explain adverse clinical events, and the paradoxical response that follows thrombolysis in patients with UA and NSTEMI .