How Are Clear Braces Different from Metal Braces?

Clear braces vs metal braces is essentially like asking is the blue Honda different than the black Honda. They are materials that are bonded to the teeth and wires are attached. The wires put pressure on the brackets, transferred to the tooth and the bone changes, allowing movement.
Clear or metal, they do the same thing.

Clear and metal braces work exactly the same way.  The only difference is that clear braces are made of plastic or ceramic and metal braces are made of stainless steel.  Clear braces generally cost more.  

There is one type of clear braces available called SimpliClear which uses clear wires as well as clear braces.  You can find an orthodontist that offers them by searching online for Simpliclear or Biomersbraces.

Clear braces (not to be mistaken as Invisalign) are still available and commonly used in virtually all orthodontic offices today.  Invisalign has gotten much better in recent years, but they still will not be able to do all corrections.  So as an alternative to the metal braces, clear braces can be used.  However, I don't personally recommend them on lower teeth for people who need overbite corrections.  These clear braces are made of ceramic, which is a very hard substance and cause excessive wear on the upper teeth if an overbite is present.  They also have been somewhat harder to remove and may fracture during treatment because the material is brittle.  Other than those  caveats, the ceramic (clear) braces work great.

It is a common misconception that clear braces move teeth slower than metal braces.  The major reason for choosing one over another is aesthetics.

One thing to keep in mind is that there are situations when clear braces would not be recommended.  This is in cases when the brackets may come into contact with teeth in the opposing arch.  Ceramic braces are more likely to abrade away healthy tooth enamel, so if you have a deep bite, your orthodontist may advise metal brackets on your lower arch.

In terms of the types of braces on the market, there is no difference between clear braces and metal braces. Both will work equally well and if used correctly, can obtain identical results.
Clear braces are made of ceramic, plastic or porcelain. Of the three, usually ceramic braces provide the best combination of esthetics, durability, and ease of removal. Often there is a slight cost difference, as the orthodontist has to pay more to stock a clear bracket system. Usually this cost is passed onto the patient. Somewhere between $300 and $500 is more than fair. Tom Cruise had the clear ceramic braces.
Historically, clear braces were plagued with problems. While they worked great for aligning teeth (just like metal braces) they were often difficult to remove and frequently fractured or damaged the tooth upon removal. With advancements in technology these issues are virtually obsolete.
Remember, advances in technology have also allowed us orthodontists to move from a bulky type of metal band that had to go all the way around the tooth to bonding (gluing) a small bracket directly to the surface of the teeth.
Your choice between metal and clear should be based on esthetics. With clear braces, someone can still "see" that you have something on your teeth, it's just not as obvious as metal braces. Invisalign is the only thing that comes close to being invisible.