I realize many of you are not aware the differences noted in the title. Here are definitions I taken out from a website. Let me know if you have any questions!!

Hope this helps~
Acrylic Nails:
There are several popular techniques that can be used to create longer better looking nail enhancements. One popular material commonly called "acrylics" is a mixture of a polymer powder and a liquid monomer (eg. ethylmethacrylate). The mixture starts to harden in 30–40 seconds after application and continues to cure to final hardness typically for more than an hour. Powder and liquid acrylics can be removed in 20 minutes using a variety of solvents (usually acetone is used). Acrylic nail powders are available in a variety of colours and can use "special effects" such as contours, sparkles and the very popular French Manicure (pink and white appearance).
Gel Nails:
Another material, commonly called "UV gel", (in correct chemical terms a polymer) resin hardens under ultraviolet light. Depending on brand these can show a broader variety of quality and properties (flexibility, strength, etc.) but may be more expensive. They generally cannot be removed by organic solvents, but only be removed by filing (or left to grow out with the natural nail).
Plastic Fake Nails
Another popular alternative to acrylic or gel preparations are Fiberglass or Silk wraps. They are done by cutting pieces of actual fiberglass or silk fabric to fit on the surface of the nail or tip and then it is sealed down with a resin or glue. These are a possible alternative for those who are allergic to chemicals used in the acrylic or gel process. Other materials can be used, as well as combinations of them. There are also temporary, cheaper flexible tips that can be quickly glued at home without help from a professional.