Interfacings are special fabrics that are used to give support to certain areas of clothing, such as collars, waistbands, pockets and cuffs.
These are usually permanently attached.
Stabilizers, on the other hand, will also support clothing fabrics as described above, but they don't have to be permanent.
Stabilizers also will temporarily support fabric while it's being sewn or embroidered on, and then removed.
They prevent fabrics from stretching out of shape.
Permanent interfacings and stabilizers can be sewn in or bonded/fused onto the fabric by ironing.
Using interfacings and stabilizers can be essential to some special projects.
For example, if you're making a handbag, it's going to need that permanent support.
Non-permanent stabilizers can be torn away or washed away in water.
Your choice of interfacings and/or stabilizers depends on how you need to use them and what type of base fabric you're going to use them on.
The weight of your fabric dictates what type of interfacing or stabilizer you need to use.
A lightweight fabric requires a lightweight stabilizer.
Medium weight stabilizers can be used on cotton or tightly woven-fabrics, as in quilting fabrics.
Wools or other heavy fabrics are going to need a heavy weight stabilizer.
If you're using a knit fabric, you will need a non-woven stabilizer with its own stretch built in.
In this case, you're going to have to match the stretch of the stabilizer to the stretch of the knit fabric, making sure that the stretch is in the same direction.
Some other uses of stabilizers:

• Using a fusible web tape makes a no-sew alternative to hemming your garment.
  
• An easy fabric scrap scarf can be made by using a long length of wash away stabilizer.
  Lay the length out on a flat surface.
  Sprinkle scraps of cloth, yarn, ribbons and threads along half of the whole length.
  Fold the other half over on top of the scraps.
  Pin both sides closely together, carefully keeping the scraps in place as much as possible.
  Now sew in straight lines closely back and forth, trying to catch all the scraps at least once.
  When you have covered the length, wash away the stabilizer, following directions on the packet.
  You have now made your own "fabric" scarf.
  
• Instead of a hoop use wash away or tear away stabilizer to stiffen the fabric while you embroider or paint on the fabric.
  Any time you hoop a fabric, you may inadvertently pull the fabric out of shape.
  
• Use wash away or tear away stabilizer or permanent interfacing as a backing for applique.
  Whole books have been written on how to create a no-sew art quilt using double-sided fusible web.
  The stabilizer is sticky on both sides, allowing you to attach it to the back of one fabric and then attach that piece to the front of the second fabric, resulting in an appliqued look.
  

Interfacings and stabilizers are the key to better-looking clothing and fabric projects.