Very simple question; why does Varys not help Lord Stark escape when he knows he is a very noble person and would serve the realm in a fair fashion, but eventually helps Lord Tyrion escape?
Esentially what BlueMoon93 said - Varys expected Ned to live. Only that Ned was not going to keep acting as Hand of the King or return to rule Winterfell, he was to be declared a traitor and sent to the Wall to join the Night's Watch, thus renouncing all claims to Winterfell. Robb would become Lord of Winterfell, provided he would come to King's Landing to swear fealty to Joffrey. That was the plan... but then Joffrey happened.
In GoT, Eddard XV, Varys tells Eddard:
"Your blood is the last thing I desire."
When Ned asks him if he will free him he says:
"I could... but will I? No. Questions would be asked, and the answers would lead back to me."
Then they talk some more and he eventually tells Eddard:
"You are being given your life back, if you'll take it."
"I want you to serve the realm. Tell the queen that you will confess your vile treason, command your son to lay down his sword, and proclaim Joffrey as the true heir. Offer to denounce Stannis and Renly as faithless usurpers. [...] She will allow you to take the black and live the rest of your days on the Wall..."
In short, because Tyrion had Jaime, and Ned had no one. It was actually Jaime (with the help of Varys) that organized an escape for him.
Varys from book is still an enigma. We don't really know what his endgame is. Though we know following fact from
ASOS, chapter 77, Tyrion
A part of conversation between Tyrion and Varys, that happened right after Jaime set him free from the Black Cells.
The eunuch was lurking in the dark of a twisting turnpike stair, garbed in a moth-eaten brown robe with a hood that hid the paleness of his face.
“You were so long, I feared that something had gone amiss,” he said when he saw Tyrion.
“Oh, no,” Tyrion assured him, in poisonous tones. “What could possibly have gone amiss?” He twisted his head back to stare up. “I sent for you during my trial.”
“I could not come. The queen had me watched, night and day. I dared not help you.”
“You’re helping me now.”
“Am I? Ah.” Varys giggled. It seemed strangely out of place in this place of cold stone and echoing darkness. “Your brother can be most persuasive.”
As for TV shows, following conversation takes part between Varys and Tyrion.
GOT, season 5, episode 1
TYRION: You never told me why you set me free.
VARYS: Your brother asked me to.
TYRION: You could have said no.
VARYS: Refuse the Kingslayer? A dangerous proposition.
TYRION: Not as dangerous as releasing me. You risked your life, your position, everything. Why? You’re not family, you owe me nothing.
VARYS: I didn’t do it for you. I did it for the Seven Kingdoms.
So, in both cases (book and TV) the difference was Jaime. It just happened so, that Jaimes feelings toward his little brother and whatever hidden agenda Varys had coincided perfectly.
He could've, but he didn't know Stark was going to die.
The plan was to have him admit that (if I remember correctly) Joffrey was the true heir of the king and then to be shown mercy. Stark was meant to live, and join the Nights Watch.
Joffrey surprised everyone when he decided to still kill Stark, and by then, it was too late.
Varys' objective was to support a promising leader and for it he'd be willing risk everything. But Ned Stark proved himself not being that leader.
Ned was a "honorable fool" (as called by Robert) and loser in the Game of Thrones. He got so many people advising him well (Varys, Renly, ... and even Littlefinger and Cersei!), and he didn't listen. He would have got himself killed again had Varys saved him. And Varys would have to pay a big price for it. Remember, he had to flee Westeros after he saved Tyrion and he became an exile. He did that for Tyrion but he knew he only gets to play that card once. When Ned was arrested, there were much more promissing players still in the game. It wasn't the right time for Varys to play that card yet.