The difference, by the numbers
Both have soda water. Only the Ice Cream Soda has chocolate syrup, half-and-half and vanilla ice cream; only the Lime Rickey has lime juice and raspberry syrup. In parts, an Ice Cream Soda is 1 chocolate syrup, ½ half-and-half and 6 soda; a Lime Rickey is ¼ lime, 1 raspberry and 5½ soda. By volume, the Ice Cream Soda is 87% mixer and juice and 13% syrup and sugar; the Lime Rickey is 80% mixer and juice, 15% syrup and sugar and 5% citrus. That makes them close cousins on the ratio map.
They are about equally strong in the glass (both 0.0% ABV), and they hold about the same alcohol (0.00 standard drinks each). The Ice Cream Soda has more calories (about 537 against 112) and more sugar (65.4 g against 27.3 g). Both are built and served in a highball.