We can't answer this in any detail without your telling us the makes and model numbers of your gear. However, an entry-level mixer that has a USB interface can usually send the two-track (stereo) mix directly to a computer without going through a separate audio interface. Mixers of this level do not have the ability to send the output of individual channels to the computer. If you want to record several separate channels to the computer for later mixing, you should do this through the audio interface, and not involve the mixer at all. Only if you have more signals to record than the interface has inputs would you look at maybe using the mixer to reduce a group of channels (e.g. the microphones on a drum kit) to a stereo pair, and then feeding this pair through two channels of the audio interface. You can't send data into the computer though both the mixer and the audio interface at the same time unless you have a Macintosh and use its aggregating inputs.
Bos did answer very well. If I may add, with the makes of the equipement, we will also be able to evaluate which of the two is the best to use.
I'm late to the thread, but have a good observation. The Allen & Heath Mixwizard4 14.2 has an aftermarket add-on available. You can add a USB 2.0 board to the mixer which will route all 16 channels to your laptop DAW. Cost about $300, available on eBay. I know it works as I added the board myself. Note, you must open up the Mixwizard yourself, screw in the board and configure it - not easy, because you have to cram it in a tight space. But, what a great feature if you're interested in more inputs of decent preamps vs fewer higher quality inputs via interface.
After reading the initial post here, my main question would be..... what is the importance of the mixer in the equation? If it is simply for preamplificatiom, most every interface on the market anymore replace that particular need for a mixer. Unless of course your mixer has a very particular set of preamps.
In fact, you don't need a mixer. Most of the time at equal price, the preamps in an interface are of equal or better quality. Nearly all audio interface have a realtime mixing capability (controled by a software). One of the best realtime mixer I've used is totalmix from RME. Any input can be sent to any output.. Very usefull... Realtime mixing is needed to make headphone mixes and the control room mix.
