As the titles says, Elizabeth has certain powers through the game which are a part of her
while the Lutece siblings need science and machines to achieve the same results
Why does she has those powers, and, is she the only one who has them, or are there others?
Potential spoilers below.
Rosalind says that it's probably because part of Elizabeth exists in another universe (her cut pinky), and thus can interfere with the tears/make new ones.
The actual quote: "What makes the girl different? I suspect it has less to do with what she is and more to do with what she's not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge."
As for the Luteces, when Fink sabotaged their tear machine to kill them, it somehow caused the siblings to be able to travel through tears freely, which is what they do throughout the game.
Needless to say – SPOILERS!
In her Voxophone log, "The Source of Her Power", Rosalind Lutece says:
What makes the girl different? I suspect is has less to do with what she is, and rather more with what she is not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge.
At the end of the game, you will learn that Booker DeWitt sold his baby (Anna) to Comstock (who was assisted by Robert and Rosalind Lutece), to pay for his gambling debts. He tried to stop Comstock from taking his baby in the last minute, but failed to do so. As the tear closed, Anna's little finger was caught in the closing tear and was cut. Rosalind Lutece says that this is how Elizabeth (which is what Comstock named Anna as) got her powers.
This answer has been shamelessly copied from my answer in another question here.
WARNING, SPOILERS ARE ABOUND!!!
SERIOUSLY, THINGS WILL BE RUINED FOREVER IF YOU KEEP READING.
OKAY, TOTALLY NOT MY FAULT IF YOU SCREWED UP THE STORY FOR YOURSELF.
YOU KNOW WHAT? I'M JUST GONNA PUT THIS ALL IN SPOILERS JUST TO BE SAFE.
Elizabeth's ability is caused by her extradimensional nature. As you discover, Comstock is infertile so he cannot sire children. Desperate for a heir, he uses the multiverse-spanning machine that the Luteces created to make a deal with a Booker from a different universe. In return for settling some kind of debt, Booker sells his daughter to Comstock, who takes her back to his universe.
Since Elizabeth is not originally from Comstock's universe, her presence destabilizes the fabric of the multiverse. In fact, her extradimensional origin is what gives her the ability to see and open tears.
