There are really two varieties of console-emulation plugin.
Cakewalks, and third party ones like Waves' NLS, which put simply emulate the changes made to audio by passing it through a high-quality mixer channel with flat eq, tend to be subtle. To really hear them make a difference it's best to have lots of tracks and busses using them. Which isn't surprising really as they are based round consoles like the SSL and Neve ones which were designed from the off to be as "pure" as the technology of the time could manage. Ironically, it's their lack of purity that's ended up making them and their sound so sought after. A bit like when CBS decided to modify the Fender amp designs to get rid of the "nasty distortion" only to find out audiences and guitarists alike actually liked and wanted the distortion.
If you want to find out what such emulations are doing, load a project with a lot of audio tracks, add the CE to each channel and bus, group all the CEs (or have them as the only effect) then bypass them all either by clicking on a group member or hitting "E" to bypass all effects.
The other kind of console emulator are plugins like Waves' Abbey Road REDD and TG12345 which, like the vintage consoles they're modelled on, have a pretty obvious and distinctive effect on audio.
Tape emulations have their uses I find, particularly if I'm trying to capture the sound of a time period when tape was used. Again, the tape recorder designers and manufacturers put a great deal of effort into making their products as hi-fi as possible so unless you crank the emulated wow and flutter and push the gain and saturation hard you won't hear them making much difference until you've quite a few channels using them.
Taking a track with a tape emulator, bouncing it, inserting another tape emulator on that track, bouncing that and repeat a couple of times is a way to emulate the old practice of repeatedly bouncing sub-mixes down because the tape deck was only four or eight channels in the first place. If you want recordings that sound like the 1960s repeated bouncing can give you the Beatles/Stones etc. sound in a DAW.
Ironic really. Back in the pre-DAW days engineers would have loved to be rid of the "negative" side of noisy consoles with less than flexible eq and tape machines with all their problems. Now we seem to want the "defects" back because they actually sound pretty good.