Buddhism has been a proselytizing religion for most of its history, meaning it actively seeks to convert people to Buddhism. Hinduism, for most of its history, has not been a proselytizing religion.

Among the original followers of Buddha were bhante and bhikkhu monks, who wandered through the country spreading the message of Buddha. They spread the early Pali canon of Buddhism (Theravada) across much of southeast Asia.

When Buddhism was first adopted by the Emperor Ashoka of the Mauryan dynasty, he sent Buddhist monks all over Asia, including Sri Lanka, China, and southeast Asia. He was very dedicated to the spread of Buddhism, setting up stupas across the country and in foreign lands, establishing monasteries. He sent his own son and daughter as missionaries to spread Buddhism.

Buddhism was also spread by mass conversions of people by their own kings. In Burma in the 11th century, king Anoratha converted his entire country to Buddhism. A century later, Jayvarman VII did the same to the Khmer people. These were conversions to Theravada Buddhism, which is the most popular form across southeast Asia.

Today, conversions are not as active. Some versions of Buddhism, like Zen, don't actively convert at all. Others, like Nichiren (a form of Mahayana Buddhism from Japan) actively proselytize and seek converts. Mass conversions are also popular in India, where entire villages of dalit Hindu castes are converted to Buddhism.

In contrast, conversions and proselytizing were absent in Hinduism. The traditional and oldest existing form of Hinduism - Sanatana Dharma - has no rituals or mechanism for conversion at all. You are free to call yourself whatever you want. You may call yourself Hindu if you like, but because Hinduism is as much a tradition as a religion, society may still not accept you as Hindu because you don't come from a Hindu tradition.

More recently, some of this is changing. It began with the Arya Samaj in the late 19th century. For the first time, they provided a mechanism - not exactly for conversion, but for return. It was during British rule, when Christian missionaries in India were converting people from Hinduism to Christianity. One focus of Arya Samaj was to bring these people back to Hinduism if possible, which they did with some moderate success.

Similar activity is still present today in modern India. There are a large number of conversions from Hinduism to Christianity/Buddhism/Islam happening today, mainly among lower castes who feel ill treated by high caste Hindus. There are several Hindu organizations that oppose such mass conversions, and actively work to return these converted Hindus back to Hinduism. It's a politically sensitive issue in India today.

But again, these people are not seeking new converts. They are trying to return ex-Hindus back to Hinduism. The only organization that I am aware of that actively seeks to convert non-Hindus to Hinduism is the ISKON (the Hare Krishnas), which was started about 50 years ago. Most of their activity is in western countries. I'm not sure how successful they are, but Hinduism remains at very low levels in western countries, and most Hindus in western countries are Indian emigres, not western converts. So it doesn't seem like ISKON has had much success.