The difference, by the numbers
Both have pineapple juice. Only the Blue Hawaii has sweet-and-sour mix, blue curaçao, vodka and white rum; only the IBA Tiki has aged rum, amaretto, hazelnut liqueur, Maraschino liqueur, passion fruit purée, lime juice and fresh ginger. In parts, a Blue Hawaii is 3 pineapple, 1 sweet-and-sour mix, ½ blue curaçao, ¾ vodka and ¾ rum; an IBA Tiki is 2 rum, ½ amaretto, 1 passion fruit, 3 pineapple and 1 lime. By volume, the Blue Hawaii is 50% mixer and juice, 25% spirit, 17% citrus and 8% liqueur; the IBA Tiki is 52% mixer and juice, 26% spirit, 13% citrus and 9% liqueur. That makes them close cousins on the ratio map.
The Blue Hawaii is the stronger drink in the glass (10.7% ABV against 8.7%), but the IBA Tiki is the bigger drink, so it holds more alcohol (1.65 standard drinks against 1.19). The IBA Tiki has more calories (about 263 against 224), but the Blue Hawaii has more sugar (23.7 g against 20.6 g). The Blue Hawaii is built and served in a highball; the IBA Tiki is shaken and served in a tiki mug.