Cross flow fans are otherwise known as Tangential Fans or Tubular Fans. It is generally tubular in shape with its length being more important than its diameter.
The air flow is in the tangential direction to the fan itself and hence the alternate name. Due to this, these fans are generally more efficient than a number of their counterparts.
Cross Flow Fans work by improving the dissipation of the heat generated by other components through circulation.
If the nearby component is already cool or the humidity is particularly high then these fans are not likely to make much of an impact.
Its advantages are that it is compact and runs without much noise amongst other things. It is mostly used in air conditioners, heating and cooling equipment to dissipate the heat generated in these components.
These fans have adjustable speed controllers and are more suited to use inside other main components.
In contrast, axial fans consist of a rotating arrangement of blades which act on the air. Usually contained within a casing, the blades used force the air to move parallel to the shaft about which the blades rotate.
The name of this specific fan comes from its propensity to blow air along the fan, and they have many modern day applications.
Within the electronics industry specifically, they are used to for cooling certain items of equipment. This is a vital cog within the whole process of constructing and transporting electrical connectors for example.
The larger varieties of axial fans are seen in mechanisms such as wind tunnels. Smaller types can be used for all manner of electronics work.
Given the size of some wind tunnels, it gives you an idea of how large certain axial fans can actually be.