The samples tend to be more midrange heavy and two dimensional whereas the synths are much wider and deeper. I know that sounds funny but these days bad musicianship can be fixed on the computer. This, I think, is another element of the warm sound we look for. Good musicianship that feels human. Sequencers and digital audio allow us to fix mistakes, many of which shouldn't be fixed in the first place.
Nothing substitutes for a nice analog desk.
Sending signals through the preamps, EQs, etc., on the mixing board does much to warm up the sound, especially at the master fader stage. The downside here is automation, as automating an analog board can be super pricey. If you wish to keep your mix virtual, I highly recommend using the insert jack available on many of the newer sound cards to insert a nice analog compressor over the final mix or to add outboard effects. It makes a major difference and can make up for some of the warmth that may be lost by the signal being relatively lower in the digital realm. By this I mean mixing down to an analog two track deck.