Ah, it's worth clarification here. Here's how we use the nomenclature:

Ticket - a message a customer sends in about an issue or question.

Bug - an issue (that can come from any source) which we track and resolve.

We just Trello for bugs (which may be reported in tickets) but we use UserVoice Helpdesk (http://www.uservoice.com/helpdesk) for tickets. Helpdesk is designed around customer communication, Trello is designed around development.

Hope that clarifies! :)