Both delegated proof-of-stake (DPoS) and Hedera's proxy staking model are motivated by the desire to allow those network participants not running node software to nevertheless use the coins they have to influence consensus.
In DPoS, such stakeholders are able to elect witnesses or delegates from a pool of candidates - only the successful candidates in this election are entrusted with acting as a consensus node. Stakeholders thus influence consensus indirectly in DPoS through their stake-weighted votes towards the election of the actual consensus nodes. As DPoS relies on human actors electing businesses into a privileged role, it has been suggested that it may be vulnerable to the standard issues of political elections - including bribery, cartel formation and poor voter participation.
In Hedera's proxy weighting model, stakeholders can directly influence consensus by proxying their stake to a node. The relative weight of that node in the virtual voting algorithm will reflect the sum of the stake the node controls and that proxied to it by various hbar holders, up to a maximum cap. Proxy stakers are incentivized to proxy to reliable nodes that will consistently earn them rewards, and the maximum cap will protect against too many people proxying to the same node.