For the same insterment (dulcimer asumed) the practle difference betwen diationic and chromatic fretting boils down to just this.
On the diationic fretting (no half frets) you have just the frets you need for a given key. The up side is theres less frets to wory with but the down side is you either have to reune or capo to play in a key diferent than the current open tuning.
On a chromatic there is no need to retune to go to another key. Its just a matter of different finger locations. You can also capo at some fret and play the same finger positions for another key. Its also possible to use argumented and diminished chords that are generally not possible on a diationic fretboard. The downside is you have all possible notes, even the ones NOT in the key you are currently playing in. That makes it possible to hit notes that are not in a given key.
I have a pair of 3 string strumstick type things I made a while back. Both are identical in every way except one has the standard diationic fretting, including the half ones. The other is fully chromatic. For me it gets confusing jumping back and forth so I generally stay with the chromatic one if Im not beating the crap outta HD.
The half frets, if you have them, are sort of a compromise between chromatic and diationic fretting. The above is a general thing but laike anything else with dulcimers, ask 10 people the same question and youre likely to get 20 answers.