It's about whose point of view t is, not pronouns.
3rd limited is from the POINT OF VIEW of the character, as if it was told by the character, only not in 1st person voice. When it's CLOSE 3rd limited, its actually told in the character's voice, so it's basically first person but with he or she instead of I. When you switch character, you're actually switching the POV to be that of the new character, and that's why you need a scene break, because you're literally switching gear.
Omniscient is told from the point of view of the all-knowing narrator. Always. There is no POV character. The narrtor knows everything from what is happening on Mars to who Mary or John or Lucky the dog or Alvin the chipmunk is, and what they are thinking. The narrator simply chooses what to tel, you, the reader. There's no switching gear. The narrator is always there.
Here are two examples. I'll use 3rd limited CLOSE vs. omniscient so you can see the starker difference (a more "neutral/distant" 3rd limited is harder to tell from omniscient in a short excerpt). Also, I'll use two characters who are 14 years old, so you can see the difference in voice (pardon the poor writing -- I'm speed-writing here to make a point):
CLOSE 3rd Limited:
Mary couldn't believe Mommy didn't let her to go to Bobby's birthday party. Not fair! She pouted, stared out the window, the crummy weather outside was just miserable. Mary wanted to scream. Then she remembered Mommy had told her she was a lady now. No screaming, and no throwing tantrums. Not fair, so not fair! Mary decided to read a book instead, but she looked all over and couldn't find it. She heard a knock on the door. She had no idea who that could be. Mommy? She opened the door and was surprised to see Bobby standing there dripping wet.
"What are you doing here?" she asked, shocked.
"I want to see you," he said. He frowned and waved his hand."Aren't you going to invite me inside?"
She thought about it for a second. "Mommy said no boys in my room."
He grinned. "But I'm not just any boy."
#
Bobby waited and waited and waited for Mary to open the door. He was wet. All wet. Hated it. What the fuck? He wasn't supposed to swear. Just couldn't help it. Ever since The Old Man had left town, Bobby had no one to give him rules or punishment. Freedom, that was what he liked. And Mary. She was cute. The door opened and the girl was right there. Sweet Jesus. She was so fucking hot. She said no boys allowed, and Bobby knew exactly what he had to say. Perfect.
"Wanna go to the shore with me?" he said. He knew she had always wanted to go there.
"Now? In the rain?" She shook her head.
"Oh, come on. It'd be fun."
She lowered her head. Then she looked straight into his eyes. "Alright."
He knew it. He knew it.
Omniscient (of the same scene):
One thing Mary could never understand was why her mother wouldn't allow her to go to Bobby's birthday party. Surely it was unfair. She pouted and stared out of her window. Outside the sky was gray and twenty miles from there, the shore was getting treacherous. It was a miserable day for the whole town, but it was even more miserable for Mary. She wanted to scream, but then she remembered what her mother had said. "You're a lady now." Her mother was serious and proud that her little girl was all grown up. And ladies don't scream or throw tantrums. Mary decided to read a book instead, but she couldn't find it -- little did she know, the book was right under her bed. She heard a knock on the door and couldn't for the life of her anticipate it to be Bobby. She thought it was her father. Meanwhile, Bobby waited outside, soaking wet after running for five minutes in the rain. He waited patiently until Mary opened the door.
"What are you doing here" she asked, puzzled.
"I want to see you," he said. Fuck, you're so beautiful, he thought. He suppressed an urge to swear in front of her. He frowned and waved his hand. "Aren't you going to invite me inside?"
She gave it a serious thought. "Mommy said no boys in my room."
"But I'm not just any boy." He had rehearsed that line the whole day. "Wanna go to the shore with me?"
"Now? In the rain?" She shook her head -- she couldn't possibly.
"Oh, come on. It'd be fun."
She thought about it again. "Alright."
He knew he had won.
OK, the writing is kind of crappy, but I hope you kind of see how the two differ. In close 3rd limited, the scenes are told from the point of view of the characters, and use their voices in the narration. When we switch, we switch the voice and the point of view, too. When we're in one character's POV (say, Mary's), we can only know what Mary can see and hear and think and feel. We don't get to know what is outside the door, or what Bobby is thinking. Same thing when we switch to Bobby's point of view. We can only hear and see, etc. what Bobby can, and know what Bobby thinks, not Mary. Even though in the same scene Bobby and Mary are present.
In omniscient, there's no switching. There's one consistent narrator and the story is told from its point of view -- it just knows Mary and Bobby and how they think and what they do and it also knows what is happening 20 miles away at the store, etc. etc. The narrator is omniscient, and it is not any of the characters and it has its own voice.
Remember, I'm doing an extreme example. In reality, even if you're writing in omniscient, it doesn't mean you should just report on EVERYTHING or what EVERY CHARACTER is thinking all the time. What it means is "you can, but probably shouldn't." It takes great skills and discipline to pull of smooth omniscient.
