Hanuman is Vayuputra, and Bheem was also Vayuputra from different yugas. Hanuman fought in treta yuga with lord Rama against the Lankan Asura army. His task as a great warrior was accomplished in that yuga. His next big task is during the Kali Yuga, to be the frontline general against the ultra-powerful demon Kali. He is one of the seven Chiranjeevis, who will defeat the demon Kali to end this adharmic yuga.
As he is from the Treta Yuga, he is more powerful than the warriors of the Dwapara Yuga and could end the war in a few days, single-handedly. But destiny has it that he shouldn't engage in the Great War and instead increase his power for the final battle with Kali. till then, he is meditating and increasing his power in the Dwapar and this Yuga.
bheem engaged in the Kurukshetra war against the Kauravas in the Dwapar Yuga. he met Hanuman during his vanavaas and gained his trust for the support during the war. Hanuman, being of the same divine birth, agreed to be present with bheem in the war, in an invisible energy form. And to protect Arjuna, he took form in arjunas flag on his chariot.
Why did he take Arjuna's place and not Bheem's is an interesting question. Bheem had vowed to vanquish all 100 kauravas through his physical strength, but Arjuna had a greater role in vanquishing the powerful astras wielded by powerful warriors such as Aswathama, Karna, Drona and Bheesma. If Arjuna was injured and out of action, the Pandava army didn't possess the astras to defeat their enemies. He needed protection from Brahmastra, Bhargavastra, Narayanastra etc. Hence, he took Arjuana's flag instead of bheemas.