First of all, let me clarify that I'm talking about non-subjective idealism with respect to the physical world (Kant and Berkeley are two examples) and about indirect realists such as (I believe) most scientists are today.
I'm looking for a list of differences that could be presented to someone who is not familiar with philosophical method and thinking. For those who are philosophically sophisticated, the differences may seem profound, but that's because philosophical thinkers think in metaphysical terms. To them, there is already a clear difference between matter and ideas. But what is that difference?
Take something as simple as a stone. There is a famous "refutation" of idealism by a Dr. Johnson who kicked a cobblestone and said 'I refute it thus'", but that doesn't actually refute any claims that idealists make. Idealists agree that there is really a stone, and if you kick it, it will really scuff your shoe; they just maintain that metaphysically the stone should be analyzed into the sense impressions that it makes.
Similarly, there is an attempt by idealists to refute realism by analyzing the disconnect between our perception of the stone and what is really out there in the world, but that doesn't affect indirect realists who are happy to acknowledge the pivotal role that our perceptual equipment plays in our perception of the stone.
We are left then, with two positions that both agree on the empirical and practical facts, arguing over details that seem a bit esoteric and fanciful. There was even a significant philosophical movement called logical positivism that made this very argument in the early 20th century.
How would you explain the significance of this distinction to someone who doesn't think the base ontological difference matters?