IMHO the most significant difference is the dynamic envelope of notes. An archetypal acoustic upright has a really strong transient at the front of notes and the decay is way quicker than the typical bass guitar. You can approximate this envelope on bass guitar with dead strings, high action, and string mutes.
Beyond that, (AFAIK) an upright has a range of different mechanism for adding energy above the fundamental: body resonance, interaction between the string and fretless fingerboard, string length, string construction etc. IMHO, an important distinction here is some of these processes vary somewhat over the range of the instrument. You might get a bass guitar to sound fairly similar to an upright in one part of the range. Getting the sound to vary like an upright as you play up and down the neck is entirely a different and far more complicated issue.
I don't have any first hand experience here, but I would not be surprised if @Podgorny is preaching the gospel here.
On a related note. Double bass players tend to pursue a grail called MBOL. The acronym means My, Bass, Only, Louder and it is virtually impossible to achieve with 100% authenticity when playing through an amp.
A piece of technology that is designed to help double bassist get a little closer to MBOL is the Audio Sprockets Tone Dexter. The Tone Dexter has to be trained. This is done by playing the bass into a mic while Tone Dexter samples the output of a contact pickup and compares it to the signal picked up by the mic. I might be talking out of A** here, but my understanding is the Tone Dexter generates a wave map that is used to process the signal from the pickup so that it sounds more like the signal picked up by the mic. I believe
@DoubleMIDI is far more qualified to explain how the Tone Dexter works; perhaps he will comment.
After the Tone Dexter is trained, you no longer need the mic. You play through the pick up and the Tone Dexter translates the output of the pickup into something more like the signal from a mic.
Supposedly you can sort of do a hack where you use a wave map generated with a double bass to make an electric upright sound more authentic. Not perfectly of course. I assume the Tone Dexter could also be applied to a bass guitar to make it sound more like a double bass. I assume the processing would be reasonably similar to what @Podgorny has described.