Somebody has already mentioned earlier that in the book, Langdon learns of Sauniere's murder from Collet when Collet goes to Langdon's hotel room. Langdon has already been asleep for a couple hours after his lecture. In the movie, Langdon goes from his lecture to a book signing--where Collet collects Langdon for a ride to the Louvre. So why the fuck does Fache think Langdon did it? Besides the fact that Bishop Aringarosa told him (which again, is a departure from the book). Langdon has a fucking alibi..."I was at a fucking lecture attended by a thousand people and I've been at this book signing ever since then. When would I have time to go to the Louvre and kill Sauniere?" In the book, because some time has passed since Langdon gave the lecture and he was awoken by Collet (and therefore doesn't have an alibi), Fache believes that it was possible Langdon could've slipped out and killed Sauniere.

There are a bunch of other key plot elements that were missing but I'm not in the mood to write them all down.

Also, and this is insignificant because it doesn't have anything to do with the plot...but when Langdon figures out where the Holy Grail is at the very end of the book, he actually goes into the Louvre where the "blood" and "chalice" meet, not outside the Louvre above the point where the two meet.

that generic old lady saying "I'm your grandmother" when it has already been established that Saunière isn't Sophie's biological grandfather, which begs the question: "Who is this woman now, her biological grandmother on her real father's or real mother's side, or else Saunière's widow, or somebody sent in from the social services agency, or what, and why should would we care? And, more importantly, should Sophie care at this point?"

This is simple to explain. Fache thinks Langdon did it because he erased the last line written in blood before he invited Langdon there, just like in the book. The line that says P.S. Find Robert Langdon. Langdon himself tells Fache that if he were laying there dying and wrote a message, it would identify the killer.

And who cares if he enters the Louvre or not at the end? It was obvious in the movie what was meant

It doesn't matter what Sauniere wrote. In the book, Fache thinks it's Langdon cause of the "PS Find Robert Langdon" thing AND because Langdon was supposedly in his hotel room sleeping with no alibi. If he went straight from the lecture to the book signing, he has an alibi.

In the movie (and the book), Langdon tells Collet that he was scheduled to meet Sauniere that night, and that Sauniere didn't show up even though he waited for him for an hour. So Langdon had to have time unaccounted for between the lecture and book signing. It's being assumed that he went from lecture to book signing, but it seems that he didn't.

A nicely acted scene is when Sophie asks him "Did you kill my grandfather?" Also what was the scene at the end with the old lady saying to Sophie "I am your grandmother" all about? I'm 24 and my grandparents are 88 and 92.

When you finish the book you'll see. It is her grandmother. I'm 29 and my grandparents are only in their 80s, so the age difference is different for everyone.

The murder start at 10:46 P.M. (book and film). Assuming Langdon was waiting for Saunière "for cocktails" (in another public place?) after his lecture and waited for an hour, he would have proceeded to the book-signing - which usually follows a lecture - around 11:46 P.M. and his public would have been waiting for his autograph a whole hour. The sensible thing to do, of course, would have been for Langdon to proceed with the book-signing while awaiting word from Saunière on his cellphone and that would still have placed him in the public eye all night. Long story short: None of this makes any sense at all... As Homer Simpson would say: "This story takes place in a faraway magic land of make-believe called France..."