Part of the decision is the finished look you want in your yard, whether it will be the more formal brick or the more casual wooden deck appearance. Brick can be laid directly on sand, which makes it a little more casual, or it can be laid on a concrete slab and mortared in place for a more lasting and formal appearance. Brick readily conforms to any curves or angles you wish to apply, as well as a great number of patterns. Wood decks are generally quite linear, running the length of the boards. Changing the direction of the run, however, can easily break up long, straight lines.
Decks go down relatively quickly.
And, of course, if you are building several feet off the ground, a deck is vastly easier than trying to construct a support system for a brick patio. Wood decks require some maintenance, such as sealing, cleaning, and repairs to damaged wood. Both brick and wood can be slippery and mossy in wet, damp areas. They are not sold to the public, but only to licensees of the maker, Bomanite Co. Retaining walls can be built out of almost anything, including bricks, fancy interlocking blocks, field rocks, old railroad ties, or even broken pieces of concrete.