Very very different. The Akbar Birbal Stories get a lot of things wrong.
Akbar is shown as a fat, old man and Birbal as a young lean man. Birbal didn't looked like the way he is shown. Birbal was 14 years older than Akbar. Birbal was born in 1528 and Akbar in 1542.
Birbal is often shown in stories cribbing about money and finding ways to get gifts and rewards from Akbar. He is sometimes also shown as a poor man before entering the Mughal court. Historically, Birbal became prosperous after getting married to a rich woman and thus never had money problems. So Birbal was economically well off even before he met Akbar.
Birbal was not the most senior or influential official in the Mughal period. He was the one closest to Akbar personally which allowed him to live close to Akbar’s palace, a privilege not accorded to other senior officials. But Birbal had a rank of 2000 mansabdari which was good enough rank but lower than the likes of Man Singh, Todarmal, Rahim Khan E Khana and Abul Fazl etc.
The biggest enemy of Birbal in these shows is shown to be Mulla Do Pyaza, a man whose historicity is contested. According to most of the historians, Mulla Do Pyaza was a character created by the Muslim clerics as a competition to the Akbar Birbal Stories as Birbal was a Hindu. Historically the biggest detractor of Birbal was Badauni who has used the choicest of abuses for Birbal in his work Muntkhab ul Tawarikh.
Also in most of the shows on Akbar Birbal, Birbal is often shown as some sort of a spy who solved the puzzles and problems in the court. Birbal came to Akbar's court after recommendation from a local ruler, and was employed as a poet. He was given the title Kavi Priya, before he was given the title Raja Birbal (named after Virvar of Betaal Pacchisi). Birbal was not a spy or judge, he was a poet and later a close advisor of Akbar.
Lastly, Akbar Birbal Stories often show Birbal saving the life of Akbar through his wit. Historically it was Akbar who saved the life of Birbal twice in expeditions. Birbal was not a military commander by training and therefore he was brutally murdered in 1586 where he was sent on expedition.