They're also lower cost, what's not to like!?!...
Well, Vce(sat) is nonzero, whereas MOSFETs can perform synchronous rectification to arbitrarily low voltage drops (simply Id * Rds(on), put enough in parallel and get Rds(on) as low as you want). This is especially important at low voltages, where the IGBT's 1-2V drop might be utterly infeasible to begin with. (I'm guessing you didn't mean ALL "all" applications, just, say at >300V or something. But if so, why overgeneralize? There are far more applications, than just whatever immediate project you're working on that brought this question to mind!)
IGBTs are also slower. A MOSFET switching in ~20ns is pretty common (given a powerful gate driver and optimized layout), making switching frequencies over 400kHz feasible, even over 1MHz for resonant operation. Part of this is the IGBT has a long tail of turn-off current, due to the bipolar current flow component, which acts like diode reverse recovery for similar reasons. This current drops off exponentially over time, and is not well captured by the turn-off time characteristic (which measures the 90-10% current, or something like that -- a large part of which can be due to majority carriers (MOS current flow)). Rather than precisely characterize the turn-off waveform, it's more common to simply give an aggregate property: turn-off energy. With a figure of some ~mJ being common, you can clearly see the losses at 100s kHz can be a huge problem!
IGBTs are also less robust. They typically show short-circuit / fault durations up to 10 or 20µs. MOSFETs typically handle five times this -- the dies are much larger. The same feature that makes IGBTs affordable (high current density) also makes them less robust.
IGBTs are also prone to avalanche failure; I think a few are available with this rating now, but by and large this is simply not available (or hasn't been, at least). If nothing else, the smaller die area means less energy handling. Not that you should be relying on avalanche in general, but it can save things in random cases. As a consequence, IGBTs tend to have higher voltage ratings (600V vs 400-500V MOSFETs; 1200V vs. 800-1000V MOSFETs -- though the somewhat fixed voltage drop means they scale somewhat better to higher voltages, or did before SuperJunction MOS were introduced).
IGBTs also don't have an intrinsic body diode, though this is a mixed bag. There are some applications where not having one is helpful; though (bare) IGBTs largely handle only -6V or so, making this a rather non-feature. (There are some available with full reverse voltage ratings, which are useful for current-sourcing and multi-level type inverters.) Co-packs (with FREDs or whatever) are readily available, and can perform better, as the FRED has less recovery than the MOSFET body diode (which tend to perform rather poorly, especially at high voltage ratings).
IGBTs are also nearly unsuitable for linear operation, not that they'd really be good candidates in the first place given their high power density -- that is, making them even more prone to 2nd breakdown, and having lower power ratings for the same V*I ratings -- though, bizarrely, there are a few IGBTs I've seen boasting a DC SOA curve(!).
Tim