As mentioned earlier, art and culture questions are factual and straightforward. This question is same too.
Briefly introduce both the religions and then jump to the reasons for Buddhism spreading out while Jainism failing to do so.
Role of royal patronage, characteristics of the religion, flexibility in practices and support from academicians and travellers will the main sub headings.
More than the discussion, your presentation will fetch you better marks. There will be lots of points to mention but in order to score better, try to frame discussion in orderly manner.
Lord Buddha was the founder of Buddhism and preached only in the area covering present day UP, Bihar, Jharkhand and Nepal border but his disciples also preached into the foreign lands. Jainism is one of the oldest religion in the world and they traced their history through a history of successors known as Tirthankaras and Mahavir was the last tirthankar. In the early age the political spread of the Bharatkhand was much larger and jainism was spread all across. It travelled outside India in places like Sri Lanka, Nepal, Afghanistan, Iran etc, with the foreign tourists visiting India at that time. But its spread was not as spectacular as that was of Buddhism. Though Jainism and Buddhism share some common features, the latter spread spread to other Asian countries while the former is confined to India.
Role of royal patronage for spreading in foreign places:
The emergence of Ashoka (273-232 B.C.) was an important turning point for Buddhism and made Buddha dhamma the basis of all his actions in the spiritual as well as temporal fields. Under his guidance, third Buddhist council was held and he send missionaries to present day Sri Lanka, Burma to propagate Buddhism.
When Greeks and Kushans established their rule over North-West India in 2nd century BC and the 1st Century BC respectively, they embraced Buddhism and popularised it in different parts of the world.
Though Jainism was also supported by some kings it was not of above scale. Ashoka, Kanishka and Harsha were pivotal in spreading of Buddhism to outside India.
Even when jainism was picked up by rulers, they immediately became ascetics and not rulers.
Different nature of the religions:
Buddhism is a missionary religion with monks and nuns (Sangha) trying to spread the message of Dharma propounded by Buddha where as Jainism is not very missionary.
Buddhism can also be easily practiced by lay people with stricter rules only for monks/nuns trying to attain Nirvana.
Jain branch of Shraman theology remained more close to it’s roots. They did not bother to preach their philosophy to anyone, because for the seekers of this branch of theology, their own emancipation was primary.
The concept of helping others never emerged in early Jainism. Their concept of helping others was to stop creating troubles for them. Thus, the entire focus of this philosophy and its seekers always remained – inwards.
Flexible practices of Buddhism:
Jains considered it a taboo to travel by water since it harmed organisms more compared to barefoot land travel. As Buddhists did not hold such a view and so were able to spread the religion to Sri Lanka, South-East Asia, China etc.
Buddhism has no strong restrictions on food for lay followers and could easily spread to communities and areas with non-vegetarian food habits.
The Belief of Jainism in cycles of birth and death of soul (reincarnation of soul) and it’s goal of liberation from this cycle -Kaivalya might not have been well received by people of other areas who did not believe in such ideas where as the main goal of Buddhism-end of dukkha (suffering, unsatisfactoriness) i.e. Nirvana was well received by people of various areas especially by kings and intelligentsia.
Role of intelligentsia:
Buddhism was also strongly supported by intelligentsia with a glorious Buddhist University (Maha Vihara) at Nalanda and other universities (Viharas) at Vikramashila, Odantipur etc. which boasts of scholars such as Acharya Nagarjuna, Aryabhata, Xuan Zang, Fa Hien etc.
This was pivotal in transmitting Buddhism to the intelligentsia of other Asian countries. Xuan Zang and Fa Hien are especially instrumental in spreading the message of Buddha to China and East Asia.
Buddhism became a prominent subject in architecture, paintings, sculptures, which were based on jataka or the previous birth stories of Buddha. This helped immensely in their popularity abroad. This aspect of aesthetics lacked in Jainism.
During colonization period, archaeologists dug up the ancient Buddhist sites, learned the Pali and Sanskrit text, and they connected the dot. This further enhanced the status of Buddhism worldwide and it started gaining traction once again.
Notwithstanding all the points mentioned, Jainism’s thoughts and philosophy are as equally admirable as that of Buddhism’s. Since the differences between both the philosophies of Jainism and Buddhism are essentially less, it is the marketing of the idea that made much of the difference between present day world population of Buddhists to that of jains. Any idea becomes a reality when it is supported by power sometimes in the form of money. This ensured Buddhism survived even when the Shankaracharya revived Hinduism all over India.