A petition, just as it is in the political arena, is a request to a sovereign to take some action. Any request made of God, is thus a petition. Prayer, on the other hand, is any communication with God.
As such, all petitions presented to God are delivered as prayer, but there are forms of prayer (glorification, confession, thanksgiving) that are not petitions.
Going to the Greek, we see it is:
παντὶ τῇ προσευχῇ καὶ τῇ δεήσει μετὰ εὐχαριστίας
παντὶ = in all things
τῇ προσευχῇ = in prayer
καὶ = and
τῇ δεήσει μετὰ εὐχαριστίας = in petition with thanksgiving.
As such one could parse the verse as "in all things .. by prayer (petition with thanksgiving) present your requests to God" as an equally accurate rendering. This seems like an instrumental construction, meaning that petitions and thanksgiving would be how the prayer is done.
Again, in short - your petition is a kind of prayer, but it is not the only kind of prayer there is.