Step 1: A dirty grey, firmly adherent membrane spanning the tonsils and pharynx in a patient with hoarseness points directly to faucial diphtheria.
Step 2: The causative organism is the Gram-positive bacillus $Corynebacterium\ diphtheriae$, whose toxin causes mucosal necrosis. The dead cells, fibrin and bacteria fuse into the classic pseudomembrane that bleeds when peeled.
Step 3: Associated features include a tender swollen neck, airway compromise, low grade fever and malaise. Toxin can later injure the heart and nerves.
Step 4: The other options are excluded because tonsillitis gives removable patchy exudate limited to the tonsils and aphthous ulcers are discrete oral erosions.
\[\boxed{\text{Diphtheria}}\]