What are the similarities and differences between facilitated diffusion and active transport?

Short answer
Facilitated diffusion is a passive process in which membrane channels mediate the transport of polar, or big molecules that are not solvable in the cell membrane. Co-transport, on the other hand, is active transport, as it depends on the electrochemical gradient of ions across the cell's membrane, particularly Na+. Because ATP or other energetic compounds are not directly involved in co-transport, it is referred to as secondary active transport. The electrochemical ion gradient is mainly established by the Na+,K+-ATPase.

Background
As commented upon by others, your question is not entirely clear and the best way to approach this question is by defining the various terminologies and physiological mechanisms used in your question:

1. Active transport needs an energy source such as ATP. For example, the Na+,K+-ATPase is an enzyme that pumps Na+ and K+ ions against their concentration gradients into the cytosol (K+) or out of the cell (Na+). Note that both Na+ and K+ are transported against their chemical gradients (and Na+ also against its electrical gradient as the cell is negative). Therefore your statement that [...] in the active process, substance x is transported against the gradient, and substance y down its electrochemical gradient is untrue, and likely the cause of your confusion.
2. Passive transport is mediated by diffusion, which typically occurs through a concentration gradient. Only small, relatively hydrophobic molecules are able to diffuse across a phospholipid bilayer at significant rates. Examples are gases (such as O2 and CO2) in the lungs, and small polar but uncharged molecules (such as H2O and ethanol) (Cooper, 2000).
3. Facilitated diffusion is an example of passive diffusion, but aided by membrane-spanning channel proteins that span the lipid bilayer. Therefore the particles (molecules or ions) do not have to dissolve in the cell membrane's hydrophobic lipid bilayer, allowing hydrophylic and larger molecules (carbohydrates, ions) to be transported into the cell. No external source of energy is needed and travel across the membrane in the direction determined by their concentration gradients and, in the case of charged molecules, by the electric potential across the membrane (Cooper, 2000).
4. Co-transport is active transport, where the energetically unfavorable transport of a particle against its electrochemical gradient is facilitated by the co-transport of a favorable one down its gradient. In other words, multiple particles are transported so as to neutralize the sum of the transport. For example, sodium-dependent glucose tranporters transport glucose into the cell against the concentration gradient (unfavorable) and Na+ is transported along with it into the cell which is a favorable transport (Haraki & Inagaki, 2012). This is an example of a symporter. An example of an antiporter is the Na+/Ca2+ antiporter found in the heart to transport Ca2+ out of the cell against a steep concentration gradient and electrical gradient, while 3 Na+ is imported into the cell to balance the unfavorable Ca2+ transport. Note that all these mentioned co-transporters and many others are dependent on the Na,K-ATPase that ultimately generates the Na+ and K+ gradients by pumping Na+ out of the cell and K+ into the cell using ATP (Freeman, 2000). This generated membrane potential works like a battery being charged, and co-transporters discharge it by using the energy stored by importing Na+. Hence, it is referred to as secondary active transport, as no direct energy input from ATP is needed, but instead it utilizes the electrochemical gradient created by the Na+,K+-ATPase.

These considerations should clarify your issues and answer your question on what co-transport exactly is. Specifically, the glucose transporter mentioned, GLUT1 as you refer to, is a symporter that mediates secondary active transport.

References
- Cooper, The Cell, 2000
- Haraki & Inagaki, J Diabetes Invest 2012; 3: 352-3
- Molecular Cell Biology, 2000

The key difference between active transport and facilitated diffusion is that the active transport occurs against the concentration gradient hence, utilizes energy to transport molecules across the membrane while the facilitated diffusion occurs along the concentration gradient hence, does not utilize energy to transport molecules across the membrane.

Molecules go in and out the cells using different transport mechanisms. Since the cells have a selectively permeable cell membrane, only some molecules or only selected molecules can go inside the cell and come out of the cell. Osmosis is the main mode of facilitating these movements in cells. Furthermore, there are two other mechanisms namely active transport and facilitated diffusion that helps molecules to move across the cell membrane. As the names suggest, active transport is an active process that utilizes ATP (energy) while the facilitated diffusion is a passive process that does not utilize ATP. That is because the active transport occurs against the concentration gradient while the facilitated diffusion occurs along the concentration gradient. However, both mechanisms take place via channel or carrier proteins located in the plasma membrane.

CONTENTS

1. Overview and Key Difference
2. What is Active Transport
3. What is Facilitated Diffusion
4. Similarities Between Active Transport and Facilitated Diffusion
5. Side by Side Comparison – Active Transport vs Facilitated Diffusion in Tabular Form
6. Summary

What is Active Transport?

Active transport is a mechanism of transporting molecules across the cell membrane by utilizing the energy produced through respiration. And, this process occurs against the concentration gradient; from a lower concentration region to a higher concentration region. Hence, unlike passive diffusion, this process demands energy. Also, either carrier proteins or channel proteins facilitate the active transport process. Furthermore, until the energy supply is available, active transport continues, and it will lead to the accumulation of necessary nutrients such as ions, glucose and amino acids inside the cell.

Figure 01: Active Transport

Besides, there are two forms of active transport; namely, they are the primary active transport and secondary active transport. The difference between primary and secondary active transport is that primary active transport utilizes ATP to uptake nutrients while secondary active transport utilizes the electrochemical gradient to uptake nutrients. Accordingly, sodium-potassium pump is a protein that involves with the primary active transport while sodium/glucose symporter is a protein that involves with secondary active transport. The uptake of glucose in the intestines of humans is an example of active transport. Another example is the uptake of mineral ions into the root hair cells of plants.

What is Facilitated Diffusion?

Facilitated diffusion or passive diffusion is the process that facilitates the uptake of nutrients across the cell membrane without utilizing energy. It uses trans-membrane proteins to transport nutrients into the cell. Since it involves integral proteins that are either carrier or channel protein, it differs from the simple diffusion process.

Figure 02: Facilitated Diffusion

Moreover, it occurs from a higher concentration region to lower concentration region along the concentration gradient. Hence, it does not require energy supply. However, unlike active transport, it does not proceed continuously. It stops at the point where the equilibrium is achieved.

What are the Similarities Between Active Transport and Facilitated Diffusion?

• Active transport and facilitated diffusion are two mechanisms that facilitate the uptake of nutrients in and out of the cells.
• Carrier proteins and channel proteins involve in these two processes.
• Also, both processes occur via the cell membrane.

What is the Difference Between Active Transport and Facilitated Diffusion?

Active transport is the process of moving molecules across the cellular membrane through the use of cellular energy. On the other hand, facilitated diffusion is the process of moving molecules across the membrane without the use of cellular energy. Therefore, this is the key difference between active transport and facilitated diffusion. Basically, active transport occurs against the concentration gradient while facilitated diffusion occurs along the concentration gradient. Thus, this is another difference between active transport and facilitated diffusion.

Furthermore, both processes help cells to uptake nutrients. But, active transport causes the accumulation of necessary nutrients inside the cells while the facilitated diffusion does not cause the accumulation of nutrients. Instead, it stops when the concentrations are equal on both sides. Therefore, this is another difference between active transport and facilitated diffusion.

The below infographic presents the difference between active transport and facilitated diffusion as a side by side comparison.

In summarizig the difference between active transport and facilitated diffusion; the facilitated diffusion is the process of transporting substances across the cell membrane with the help of carrier or channel proteins. It does not utilize cellular energy. On the other hand, another mechanism called active transport utilizes the cellular energy in order to transport nutrients, especially ions across the membrane. Hence, this is the key difference between active transport and facilitated diffusion.

Reference:

1.“Active Transport.” Khan Academy, Khan Academy. Available here
2. Editors. “Facilitated Diffusion (Transport): Definition & Examples.” Biology Dictionary, Biology Dictionary, 28 Apr. 2017. Available here  

Image Courtesy:

1.”0308 Sodium Potassium Pump”By OpenStax  (CC BY 4.0) via Commons Wikimedia  
2.”Scheme facilitated diffusion in cell membrane-en”By LadyofHats Mariana Ruiz Villarreal – Own work. (Public Domain) via Commons Wikimedia