Even if the tear only gets 1/3rd charged, it still is giving 350+333 = 683 mana for 1000 gold, plus the mana regen which is kind of helpful early game.
This is still pretty efficient and since mana is just as good as AP offensively (for the gold cost), this puts Tear as a quite efficient item. If it gets fully charged it is giving 1350mana for 1000 gold which is hugely efficient (2700g of mana for 1k).
The difference between tear and say deathcap is that the tear is far better in letting you spam early and midgame due to all that mana. The large amounts of mana also help when you convert to an archangels staff.
You don't HAVE to start with a tear though. In fact, you might go with a Catalyst instead especially bottom lane. That makes you much tougher and you get health as well as mana regen. Get the tear (and boots) on your first Back, then move on to your other usual items.
You could also get away without the tear, as long as you have something to give you early and midgame mana regen. You could start Catalyst, then instead of tear, you grab your glacial shroud or upgrade to an early Odins, or even take Rod of Ages.
A Deathcap nicely multiplies itself as well as your Rod and Abyssal Scepter AP.
So TL;DR - You dont HAVE to take tear, but its a strong item and worth taking.