It's true that LDCMs are more sensitive, but that's in practise not such a big concern – good SDCMs already have plenty of headroom, usually the noise floor is well below any ambient noise even for chamber music room applications. Of course, the diaphragm mustn't be too small, cheap mics with less than a centimetre usually won't do. But they still beat anything you could practically achieve with dynamic microphones. The commercially-available microphones with the highest sensitivity are actually HF-biased small diaphragm mics such as the Sennheiser MKH 40 P48.
One obvious difference is a mechanical aspect: condenser mics are always more vulnarable than dynamic mics, but LD ones are particularly fragile. Combined with their clunky heaviness, this already makes me hesitate to use them in any live setting.
But most important is the sound. I like SDC mics a lot because they cover the entire spectrum quite linearly, behave very predictable, have excellent phase response and can be aligned any way you want. LDCMs usually have already quite a shaped frequency response. Nothing dramatical and often it sounds just great, but if you like to start with all-linear it's not really nice. The transient and phase characteristics are still very Dirac-like – always much cleaner than dynamic mics – yet for percussive instruments and even guitars I sometimes feel it already lacks some of the precision that SDCMs offer.(Though actually I doubt the scientific significance of that observation, because even studio headphones can't really hope to match the phase response of a LDCM.)
Where the diaphragm size starts to matter really a lot is when doing close-field recording, like most vocal recording nowadays. With SD microphones or omnidirectional microphones, distance doesn't make to much of a difference sound-wise, you just get a higher amount of direct signal compared to the room components. But with the commonly used large-diaphragm cardioid microphones, there's quite a strong proximity effect; basically you get a natural bass boost without making the response muddy. This is highly sought-after for vocal recording and can sometimes be benefitial for drums etc., but most of the time it just adds frequency components you don't need and don't really want. When recording guitars with LDCMs, I generally leave a bit of space so the proximity effect won't be too pronounced. Given good room acoustics, this can sound very nice, but rather more often I opt for SDCMs and close miking, resulting in a rich yet clear and direct sound. You can still add room mics, internal mics, perhaps even piezo pickups or contact microphones for extra sound components, their influence will be much more controllable in the final mix.