Pan-(romantic/sexual) can be defined as having an "attraction to people of any gender or to people regardless of gender."
Omni-(romantic/sexual) can be defined as "the attraction to people of all sexes and gender identities, wherein gender plays a role in one's attraction." Even though omni- individuals recognize the gender of those to whom they feel an attraction, it may not play a huge factor in their choice of partner.
People who identify as bisexual may be pansexual, but not necessarily. All pansexual folks can be classified as bisexual (being attracted to “all” fits into the description of “2 or more”), but not all bisexual folks are pansexual (not all bisexual folks are attracted to all genders).
Those who identify as pansexual may feel attraction to people without noticing their gender, while omnisexual people recognize the gender of potential partners.
Some people may prefer to identify as bisexual even if they may be pansexual simply because the term “bisexual” is more commonly recognized. Omni- and Pan- also implies an attraction to all genders.
These definitions are not meant to be restrictive, and people can choose which one best describes themselves. Some might be used interchangeably, while others are used uniquely, since one's knowledge and familiarity with each term can affect their usage. I myself am still exploring ways to describe myself, and have found this research insightful.
From left to right: The bi- flag, the pan- flag, and the omni- flag.