Firstly, a remark- I am not expecting to make a live drum set sound just like a recorded drum set. Also, let's restrict the scope of this question to certain pop/rock/funk/metal/country types of drum sounds... Examples will be provided.
If you listen to a lot of recorded music or live music in large venues that warrant a miked up drum kit, you might notice that the drums are frequently processed to fill a certain kind of sonic space- for example, the kick needs a strong and tight low end, the snare needs to be punchy with a defined "crack" attack, the cymbals are crisp and not too harsh, and everything is balanced in a musical way.
Examples, since descriptions only get us so far:
Now if you were listening to same drum kit from any of these recordings live in person (or if it were recorded with a single overhead mic), with no electronic enhancements, there would probably be a number of differences, maybe even "problems"-
The snare tends to be way louder than all the other drums, with a lot of mid range harshness that can be overbearing. The cymbals can easily overpower everything else and can sound quite harsh compared to the nice crispy studio tracks. The overall sound doesn't have nearly the amount of low end as the recording- recorded toms can be quite bassy ("like cannons", even). On top of that. you get even farther from the recorded sound once you tell the drummer to "keep it down", which is often, as an unmiked drum kit (played in these styles) can be loud for even a 200-person outdoor venue.
More illustrations- here's an example of what my drum kit sounds like*
Since this is recorded, I can make it sound a bit fatter with an additional mic and some heavy-handed post processing
But the live drum kit doesn't sound like this at all, and I can't post process it in real life.
Now, obviously a live acoustic drum kit fills the same musical role as a recorded drum kit. So, in a venue that's too small for a fully miked and processed drum sound, how could we engineer the drums to fill the same space (from a sound engineer's perspective) as these recorded drum kits, short of abandoning acoustic drums completely?
*This recording was done with a single vocal mic placed overhead, and then eq'ed with the opposite eq curve of the "frequency response" shown in the microphone's user manual. I would love to have a recording done with a calibrated measurement microphone here, but I can't find one, so this is what I have.