Neoliberalism is an approach which is best attributed to free trade agreements and large levels of corporate and government investment being offered to underdeveloped countries. It is an approach favoured by most Western nations such as the USA and UK.
The promotion of free trade via the removal of protectionist barriers would encourage development of the world's poorest countries by making it easier for them to sell their primary products in the developed world. This can be successful, as shown by the removal of the USA cotton subsidies, which impacted positively on Mali, who was then able to reduce it's trade defecit and debt levels. This then allowed for investment in healthcare and education and levels of development increased.
However, free trade can be detrimental to development, as shown by French Guiana. Here, the removal of protectionist barriers led to the decline in the cocoa industry, as farmers failed to cope with intense international competition.
Grassroots strategies are usually small community projects which focus on basic needs (water, shelter, food, equality). They are favoured by NGOs such as Water Aid and Oxfam.
These strategies increase the quality of life for communities which guarantees a healthy and more able workforce to generate a higher standard of living (a healthy workforce means more money earnt which means more disposible income). Furthermore, they are non-eurocentric schemes which promote national pride and identity. Successful schemes include "the Girl Effect"'s female education campaign in Rwanda.
However these schemes are small by nature, they affect individual communities but have limited funds and they cannot reach everyone who needs their help. Furthermore, they do not tackle bigger issues such as government corruption, which is a major barrier to development.
So which is best? Grassroots is obviously the most sustainable, but smaller scale. Neoliberalism technically is a good strategy, but only when implemented properly (with consultation with the local community). The best way of closing the Development Gap is changes to trade barriers and political reforms, combined with the Neoliberal projects (hopefully with additional funding).