To simplify a complex situation, the difference is because the dynamics of India-Pakistan relations are defined through the lens of the Partition in 1947, while India-Bangladesh relations are based on the Liberation War of 1971.

India-Pakistan: Post-1947 relations are characterized by periods of war, insurgencies, and attempted reconciliation. Major conflicts include the wars in 1947, 1965, 1971, and one after 1998, with the epitome being the nuclear arms race in the 1970s. Cultural relations followed a similar pattern of dislike. This is visible in affairs such as the Wagah border ceremony, a flag lowering ceremony characterized by aggressive posturing from both the Pakistani and Indian soldiers; and cultural artifacts, such as the Bollywood (Hindi film industry in India) movie Border (1997), which dramatizes the Battle of Longewala in the western sector of the 1971 war; and cricket — the book A Corner of a Foreign Field by Ramachandra Guha (2003, Pan Macmillan) has two chapters that show how the sport interacted with nationalist feelings in both nations.

Pakistan-Bangladesh: Pakistan in 1947 was comprised of two units, Western and Eastern Pakistan, which were separated geographically, economically, and culturally. National integration, political, and economic policies tended to favor Western Pakistan exclusively, such as in the choice of Urdu as the national language till 1954. These tensions resulted in the growth of the political movements that asserted Bengali identity, such as the Language Movement, which ultimately boiled over in 1971.

In the December 1970 general elections, the Awami League, led by Mujibur Rahman and based in Eastern Pakistan, won a numerical majority but was prevented from forming a government by Yahya Khan, the President of Pakistan, and the Pakistan Peoples Party in Western Pakistan. A strike was declared by the Awami League, and the government essentially lost control over the region. On the 25^th of March 1971, Operation Searchlight was launched to suppress secessionist elements, and Rahman was arrested. Violence erupted, with the Pakistani military ultimately carrying out what US diplomat Archer Blood described as a "selective genocide".

India-Bangladesh: Indira Gandhi, the Prime Minister of India, decided to intervene militarily in the conflict. This followed diplomatic efforts to call international attention to the conflict. However, Pakistan at the time was a major ally of the US in the region, and the Nixon administration favored the existing government — Task Force 74, a carrier group from the Seventh Fleet, was dispatched to the Bay of Bengal at the height of the conflict, posturing for Pakistani forces. The reasons cited for Indian involvement included humanitarian aid, complications caused by the influx of a few million Eastern Pakistani refugees, and/or protection of Indian geopolitical aims against Pakistan. Hostilities began with pre-emptive air strikes and shelling by Pakistani forces on the 3^rd of December. The Indian Army responded on both fronts, acting defensively in the west, and targeting Dhaka, the chief urban center of Eastern Pakistan, in the east. Hostilities ended with the signing of the Instrument of Surrender in Dhaka on the 16^th of December. Given India's role in the war, Pakistan's hostility to Bangladesh post-1971, and Bangladesh's need for economic support and diplomatic recognition, it was unavoidable that Rahman, appointed the President of Bangladesh, would foster good relations with India.

It should be noted that post-1971 relations between the two nations have not always been friendly. The influx of Bangladeshi refugees led to the Assam Movement in the early 1980s, which included the Nellie Massacre in 1983. Issues over sharing the waters of the Ganges were resolved only in 1996 with a bilateral treaty, though tensions continue.

Sources: The 1993 book India's Partition: Process, Strategy and Mobilization (ed. Mushirul Hasan) provides a collection of multiple perspectives on the Partition; and the events and developments that led to it. A Revisit to the Indian Role in the Bangladesh Liberation War by Zaglul Haider (2009, Journal of Asian and African Studies 44/5) provides an overview of the role of India in the 1971 war (though his conclusions skews solely towards realpolitik). Srinath Raghavan's 1971 (2013, Permanent Black) has a good overview of existing narratives of the war. The Colonel Who Would Not Repent by Salil Tripathi (2016, Yale University Press) contains several interviews of participants and witnesses of the war. Sarmila Bose, who wrote Anatomy of Violence (2005, Economic and Political Weekly 40) and Dead Reckoning (2011, Hurst Publishers) is a controversial author who holds that 1971 was solely the result of ethno-linguistic “Bengali” nationalism — see Flying Blind by Naeem Mohaiemen (2011, Economic and Political Weekly 46) for a critique of her position. The Blood Telegram by Gary Bass (2013, Alfred A. Knopf) is a good overview of the actions of the Nixon administration during the war. This can be supplemented by the 2015 study done by Mohammad D Hossain, Manufacturing consent (Journalism 16/4) which analyzed the framing of the 1971 war in the New York Times and the Times (London), and found that emphasis was placed on a neutral narrative of the war as a military conflict.

Edit: fixed grammar and dating; added some more sources; corrected the relationship between Yahya Khan and the Pakistan Peoples Party.