The difference
The Between the Sheets is a Sidecar with white rum added: Cognac, rum and triple sec in equal parts, with lemon a smaller share. The extra spirit makes it the bigger drink, at about two standard drinks to the Sidecar’s one and a half.
By the numbers
Both have Cognac, triple sec and lemon juice. Only the Between the Sheets has white rum. In parts, a Between the Sheets is 1 rum, 1 brandy, 1 triple sec and ¾ lemon; a Sidecar is 2½ brandy, 1 triple sec and 1 lemon. By volume, the Between the Sheets is 55% spirit, 27% liqueur and 18% citrus; the Sidecar is 56% spirit, 22% liqueur and 22% citrus.
The Between the Sheets is the stronger drink in the glass (18.7% ABV against 17.8%) and holds more alcohol (2.03 standard drinks against 1.58). The Between the Sheets has more calories (about 233 against 179) and more sugar (8.0 g against 5.5 g). Both are shaken and served in a coupe.