
Raspberry syrup is raspberries cooked down with sugar. Unlike a plain sugar syrup it brings acidity of its own along with the sweetness, and a deep pink-red that carries into the glass.
Berry acidity under the sugar is what separates it from a plain syrup, and most red-berry syrups are sweeter and flatter. Purée with syrup rebuilds it and lets you set the sweetness, since bar purées come unsweetened.
Grenadine
Pomegranate does the same two jobs, sweetening a drink and turning it red. It is sweeter than raspberry, though, and tannic instead of tart, so it sits oddly in a raspberry drink.
Raspberry liqueur (framboise)
Around 16–20% ABV with roughly half the sugar of a syrup, and start at half the volume the recipe gives and make the sweetness up with simple syrup.
Homemade syrup is sugar and fruit juice, and two weeks in the fridge is about as long as it lasts. It warns you on the way out, clouding first and then growing mold. A sealed commercial bottle lasts months and moves to the fridge once opened.