That is a great question! I only started playing around with the Mandolin in December, when my son got one for Christmas. (Should have done it ages ago).
Basic differences (which I hope you know some of):
1. Fretted
2. Scale is a bit longer (like 13-3/4" or so vs 13" for violin)
3. 4 courses of 2 strings and you pluck or strum it
4. Strings are very tight.
Now, they are totally different instruments. Sound production of a bowed instrument is unique--you don`t get that on a plucked instrument.
However, the "standard" tuning is the same as a violin, and so provided you can get used to fingering the neck in a lute position (rather than like a violin) then it is quite natural to the violinist in terms of finding the notes. The slightly longer scale sometimes trips me up but not usually--since it is fretted, if you land a bit "flat" it does not matter :-)
The hardest parts are (for me):
1. picking the precise string/strings to be plucked
2. pressing strongly or precisely enough to get a nice sound, without damping a neighboring string or digging ruts in my fingertips.
It is very helpful to focus on putting the pad of the thumb on the back of the neck (opposite the "front"--the fingerboard). The thumb is then in direct opposition to the fingers, and you can get a good reach and good pressure. If you wrap your thumb up and around, you cannot make good finger pressure.
Also make the fingers come in fairly vertical--helps with the pressure and preventing damping--but it is painful on the fingers at first.
I also find that keeping a Mando in tune is a challenge--and the frets are a compromise in terms of intonation. You can hear the "sharpness" of a major third for instance--whereas on a violin you can play it pure.
(Major third is a frequency ratio of 1.25:1, or the second diatonic note above the tonic--the third note in the series, which for fretted instruments is a frequency ratio of 1.2599:1)
Another important difference to the violin:
Mandolins are "short scale" instruments and so the intonation is very sensitive to string gage, bridge height, saddle positions etc. Check the 12th fret vs octave harmonic and be sure you have a close match--otherwise move the bridge or change the action height. Violins do not have this problem
The best lesson is to take the mandolin chords and transfer them back to the violin--then you really get some good cross-pollination (my son taught me that idea!).
regards,
Bill