Why does Austen call Elizabeth by her first name and mr darcy by his last?

Names were used a little differently in those days.

Calling a man by first name implied a close familial level of intimacy, so the only people in the book likely to address Darcy as Fitzwilliam would have been Georgiana and possibly Elizabeth, after they were married. A friendly level of intimacy would be shown by addressing him by last name, and we see Bingley do this often. Normally, a formal address was used, which combined title and last name, as in "Mr. Darcy." Notice that Mrs. Bennet still addresses her husband as Mr. Bennet, and Bingley's sisters address Darcy as Mr. Darcy.

For a woman, the use of first name only was for close familial intimacy, as between Elizabeth and Jane, or as Mr. Bennet would address them. Jane was called Miss Bennet because she was the oldest daughter. The other bennet girls were addressed the same way, if there was no chance of confusion, but usually called by title and first and last name, formally, or by title and first name, informally.

The author and reader are on intimate terms with Lizzy, and slightly less intimate terms with Darcy.