NCI and NHLBI are the two programs within NIH. They are separate programs. NCI has many more training slots (6-10 or so I believe) than compared to NHLBI (they take around 3 annually).
NCI is focused on solid. NHLBI is focused on heme. They each have faculties that is compartmentalized to either NCI vs. NHLBI. Fellows can work with faculties from either side - however, i feel that it may be unspoken that they would prefer fellows to work within their compartments e.g NCI vs. NHLBI. Fellows in either programs have the opportunity to double board, just that the site of rotation is different. NCI fellows go to Georgetown for run of the mill cancer exposure. NHLBI fellows go to Washington Hospital. They also do a acute leukemia month in Hopkins. Obviously, NHLBI fellows get more heme exposure while NCI get more solid exposure within their 18 months of clinical training. Both NCI and NHLBI have their own transplant service - however, i believe NCI mostly do autos (which is not strictly a transplant e,g immunotherapy, since it is more of a stem cell rescue rather than achieving GVL effect) while NHLBI does allo/haplo/cords and they also perform experimental transplants like haplo-cords leading to several donor chimerism in the early transplant period.
NHLBI faculty has traditionally focused on MDS, aplastic anemia, bone marrow failure and BMT. They also do basic science on hematopoiesis. As a result, trainees who come out from the NHLBI commonly do BMT, MDS, aplastic anemia and bone marrow failure syndromes within academia.
NCI. I think they loss several big names including a myeloma person I believe. They are strong in lymphoma. The last I know, they have patchy expertise within solid tumor. Perhaps someone can provide an update.