I don't know which demo you mean exactly, but 'analog' qualities can be achieved in a number of simple ways in the digital realm (without even getting into advanced modeling). A few of them:

- Focusing on upper harmonic content. Starting with a saw wave gets you far here, as it starts with _all_ the odd and even harmonics which you can then gently trim away. In particular, even harmonics in the mid range are often described as 'warm.'

- Using resonant filters. These too contribute to a sense of richness and warmth, especially if the resonant peak is closer to the mid range.

- Adding a sub oscillator below the primary one to give it a subtle low hum.

- Adding more oscillators and detuning them slightly with respect to each other for stereo width and play in the harmonics.

- Modulating pitch and filter parameters with slow, gentle LFOs.