Tradeoffs when choosing the op-amp package: quad vs dual vs single
With no exceptions that I can think of, using four op-amps requires that all the op-amps be of the same type. In many cases this means that some of the op-amps will not be optimal in some way. The type you would pick for an output circuit might have high slew rate, high power supply voltage capability, high drive current and be tolerant of capacitive loads, however those characteristics might be unimportant for the front end which requires (say) very low noise, offset voltage and low input bias current. So chances are good something in the performance or cost is being compromised compared to using more part types. There are many types of amplifiers that all co-exist in the market for good reasons.
Crosstalk, which others have mentioned, can take place even at DC in The form of slight shifts in offset voltage of one op-amp with the output voltage of the other- possible significant when they are operating at very different levels. Thermal crosstalk also as mentioned is also possible and can cause distortion or intermodulation.
Layout is often easier with singles or duals compared to quads, in my experience. There is little advantage in board space with quads over duals, however quads can be cheaper per op-amp (the ubiquitous LM324 is typically the same price approximately for the dual and quad so you are almost getting two op-amps for free - except board space and power consumption). Higher performance op-amps usually are not like that.
Modern op-amps include many types that have very limited power supply voltage- perhaps as low as 5V (+/- 2.5V) maximum. Parts with similar performance that can handle +/-15V are much more expensive or unavailable. So using a mix of supply voltages may make sense, which requires different packages.
I can't really answer the question about what parameters are important - every number and graph and paragraph in the data sheet and any application notes is important in one application or another, as well as other things which may be unstated or hidden on the datasheet. For example, a cheap LM324 is used in an application where one amplifier is used as a comparator, but the requirement changes and higher performance is required, so the designer substitutes a more precise type, unfortunately the substitute is a wonderful op-amp but does unfortunate things with a large differential input voltage causing much gnashing of teeth and loss of hair. The information is hidden in a block diagram on page 27 of the datasheet where series resistors and back-to-back diodes are shown but not in any performance number since they are not specified under such conditions.
The thermal effects of having multiple op-amps in a single package may become a concern, as the op-amps output voltage changes so does it's thermal dissipation. This thermal wave will propagate across the chip causing imbalances in the op-amps input. Therefore if you are trying to use a high speed op-amp to drive a low impedance load your thermal dissipation could around 30C difference
Furthermore many will share bias circuits internally and improper use of op-amps may cause these circuits to fail for all of the op-amps not just one.
Crosstalk, in two forms, both electrical and thermal. Thermal from the wave propagating across the chip will look like low frequency feedback across the input. Electrically as supply pins are shared so a large load in a single section will cause a drop across the bond wires.
When using a quad chip or even a dual chip you risk limiting yourself. As all op-amps are identical and therefore you must pick one style/feature whether that be aimed at capacitive loads, slew rate, hpv capability etc. You may be better having many single different op-amps rather than one quad for this reason.
Hope this helps :)
Crosstalk can be a problem in some applications.For Audio stuff which I dont do all the time I use seperate opamps for each channel because the crosstalk can be distorted so its not just stereo seperation . In other words the crosstalk spec appears more than adequate compared to the stereo seperation of normal program material but such an amp sounds ghastley, only to be cured by using seperate chips for each channel.