The trick here is to pick the sign that is present in one entity but absent in the other, rather than a sign shared by both.
Think mechanistically. In mechanical obstruction the gut is healthy and keeps trying to push contents past a physical barrier, producing waves of contraction. Clinically this gives $colicky$ pain that comes and goes with each peristaltic wave, accompanied by loud, high-pitched, tinkling bowel sounds.
In paralytic ileus the smooth muscle itself has stopped working (post-operative, electrolyte disturbance, peritonitis). With no peristalsis there is no cramping - the abdomen distends quietly and bowel sounds are absent.
Testing each option:
$\bullet$ Abdominal distension to both.
$\bullet$ Vomiting to both.
$\bullet$ Tachycardia to both (fluid loss / sepsis).
$\bullet$ Colicky pain to only mechanical obstruction, because it requires active peristalsis.
\[\boxed{\text{Colicky pain is the differentiating feature}}\]