My guess that you might be missing phrasing and articulation:
- legato vs. detached vs. staccato notes, which in turn becomes (an aspect) of phrasing across multiple notes.
- controlling the volume of the instrument, both between different notes, and as a note sustains, so that you can put accents on different beats or provide articulation on the notes.
- the slight variations in rhythms (e.g. long-short-short for a waltz) for different types of tunes
Peter Laban posted a link to this article on playing the pipes in another thread years ago, some of which applies
directly to concertina, some of which only partly so, but I found it to be a good discussion that got me thinking about these aspects
I usually consider it good practice to initially work on this while playing scales: play a scale fully legato,
play a scale in legato pairs c-d e-f g-a ...etc. Play soft, play loud, play alternating nodes loud-soft etc.
You can do the same kind of treatment to tune snippets, but then you have to think about two things at once.