Jira was originally designed back in 2003 for bug and issue tracking. Its feature set has expanded to encompass agile project management, service desk ticket tracking etc.
Trello features JIRA like Kanban boards that allows users to collaborate, organize and prioritize their projects. In a nutshell, a Trello Board helps organize any project by breaking down big tasks into bite sized pieces with lists and cards. A Trello board is made up of lists (such as To Do, Doing, and Done). Each individual task in a list is represented by a card. Tasks can be easily delegated by simply dragging and dropping their avatars onto cards.