Some of the differences you would find in an Old Believer Liturgy is the amount they say "Lord have mercy". Such as at the dismissal they would say "Lord have mercy, Lord have mercy, Father Bless" instead of three and they will also only say Alleluia twice after reading Psalms. In the Litany of Fervent Supplication, either one "Lord have mercy" is said, or three, and then twelve for example:
"O Lord Almighty, God of our Fathers, we pray Thee: Lord have mercy
Furthermore, we pray for the prosperity, peace, tranquility, order and remission of sins of all Orthodox Christiandom. Let us say: Lord have mercy (Twelve times)
"Furthermore, we pray for all who serve or have served in this holy temple and for their health and salvation. Lord have mercy (Thrice)"
Prostrations are done more often during the Liturgy whether it is Sunday or not for instance they always do a prostration after the prayer "It is truly meet" and "Shine, shine, New Jerusalem" during Bright Week; "It is truly meet" is finished with a prostration all throughout the year. When prostrations are done, Old Believers use a small prayer rug called a podruchnik which is used to keep the hands clean during prayer after doing prostrations. There is more order in the services of Old Believers. There are appointed times when to do the Sign of the Cross, bow, and do a prostration so that everyone does them in unison so they don't do the Cross and bow whenever they feel like.
You can read about the different practices of the Old Rite at the Church of the Nativity's website: http://churchofthenativity.net/index.php?option=com_content&task=category§ionid=6&id=22&Itemid=48
Church of the Nativity is an Old Rite Church under ROCOR and they have a list of practices like conduct during services and entering a church. They publish a really nice prayer book that I would highly recommend called the Old Orthodox Prayer Book
which includes Vespers, Matins, Hours, and Liturgy.
Here are also rules of piety: http://churchofthenativity.net/index.php?option=com_content&task=category§ionid=5&id=15&Itemid=52