Is it true that there would be no snow without dust?

We need to have dust to form a snowflake. The ice crystals must attach to a dust or pollen (also called aerosols). When enough ice crystals attach to the one with dust in it and make it heavy enough, it falls down.

There are two types of Nucleation:

1. Homogenous nucleation
2. Heterogeneous nucleation

Homogenous nucleation, occurs when ice forms without any predefined nucleation site. Pure water will freeze at approximately -39°C in the absence of nucleation sites. Homogenous nucleation is rare due to the existence of completely pure water.

Heterogeneous nucleation, which occurs when ice begins to form around a nucleation site, such as a physical disturbance, an impurity such as dust or pollen. This type of nucleation takes place more easily due to the presence of an impurity in the atmosphere.

The SnowFlake may be induced without the presence of tiny dust. Now we know that, heterogeneous SnowFlake will form instantly with the help of microscopic particles at temperature higher than the homogenous SnowFlakes.

There are two ingredients necessary in the atmospherics' air that ice crystals from and these crystals aggregate to snow. I agree to @creepy-creature there is heterogeneous nucleation favored to homogeneous nucleation. But snow is a common name to atmospheric crystals that appears on earths atmosphere. The earths atmosphere does not allow very much for homogeneous nucleation at all. The set of conditions is therefore by experience different.

In the atmosphere not dust particles or the semen of plants are responsible for nucleation but chemical constituents. Snow flakes built up out of chain of nucleation steps from frozen water. These steps are pure physical steps of higher nucleated aggregations. The initial step is efficient in the present of activating chemicals.

Three properties of the atmosphere are sufficient for efficient nucleation rates:

1. Sufficient high atmospheric humidity (water partial pressure).
2. A temperature below 0°C.
3. Freezing initiating snow germs, chemical particles.

The start are supercooled water drops. These have a temperature below 0°C. In the vicinity of these super cool water drops the snow crystals form. The aggregation step water steam, heating snowflakes and cold snowflakes hit and chain up more and more. This chaining process continues on and on in the clouds.

A snowflake can contain up to many million ice crystals. The fall to earth starts if the snowflake either gain momentum by winds or are to heavy to remain in the snow feeding rang in the clouds.

The form of the snow crystals depends on the parameter temperature and air humidity. At temperatures between -6°C to -10C pillars are formed, between -12 °C tiles and starting below -12°C stars. The conditions during snowflake development may differ from place to place of aggregation. Therefore each snowflake is expected to be individual and single.

The size of the snowflakes that built up the new snow on the ground depends on the temperature on the ground and the types of the grounds. From above -5°C on bigger snowflakes. At lower temperatures, the snowflake gets smaller.

In snow machine aerosols or chemical germs do the task. High dispensed water drops of the size of 1 micrometer are mixed in an aerodynamic jet together. The properties of such mixtures are snows that remain snow even at temperatures above 0°C. An example is $NH_4NO_3$, called Ammoninanitrate. This is an artificial fertilizer

The modern SERPs of google for example offer a bunch of answer alternative and answers to such kind of questions. It is possible to get definitions both for snow and for dust. Dust is a much wider class than aerosols. Some dust may be aerosol emitter and be classified as aerosol sources but not vice versa.