By definition, Polyamory is the practice of, or desire for, intimate relationships with more than one partner, with the informed consent of all partners involved. It has been described as "consensual, ethical, and responsible non-monogamy".
“The fundamental philosophy of polyamory is that sexual love shouldn’t be confined to the strictures of monogamy, but expressed freely and fully,” ~ Susan Winter
An open relationship, also known as a non-exclusive relationship, is an intimate relationship that is sexually non-monogamous.
In a polyamorous relationship, each part of the relationship is faithful to that group and does not date outside the group, unless with permission from the rest. These cases occur during instances where one person cannot meet all the sexual needs of more than one partner and one partner decides to source sexual satisfaction from somebody else, that is not part of the domestic group. All persons involved in the polyamorous group must consent to this or the partner is basically cheating, which is as much a crime as it is in monogamous relationships.
In open relationships, however, there is honesty maintained between both partners but there is little need for discussion about other sexual encounters, unless in cases that involve sexual health or if it's important for the other person to know. Or in some cases, if the external party is joining the couple for sex. The rules are not as stiff as in polyamorous relationships because the agreement is mostly for sexual needs, hence the third party needs not be involved in the domestic arrangements of the couple in the open relationship. The third-party (or any number really) serves one purpose only; sex, and outside of sexual activities, there is no intimate relationship or any form of relationship at all, in most.
Both kinds of relationships demand a ton of communication and honesty though, as they involve letting new people into private spaces where relationships have already been formed. Some people go from monogamy to polyamory, so it takes a lot of communication and complete opening up, to convince the partner that's already there, that you're not saying they're not enough or that they don't satisfy you. Jealousy is still a possibility in both because people tend to see things differently when they're looking from the outside and a partner can feel you're giving your other partner more attention than them. They both take a lot of work and contrary to popular opinion, are not as easy as they seem. It's not just an opportunity to have sex with more people, there are feelings, time, sacrifices and plenty of work involved, to make sure you're satisfying your partners as well as yourself.
Monogamous relationships are way easier if you're not prepared for the work and dedication, but for those who can manage it, it can be an answer to meeting several of your needs, if you have the ability to care for more than one person, or in the case of open relationships, explore without leaving your partner behind.