Mixer may not be the same usually as an audio interface. An audio interface usually has a combination connector (XLR - Neutrick combination connector) for microphone, or synth, or guitar, so yes, it includes a line in, a XLR connector, and a regular 1/4" guitar cord plug in connector.
An audio interface inputs the sound into the recording program of the computer and also plays back the recorded sounds (recording, movie, utube whatever) from the computer same as a sound card would, but has professional connectors that may include XLR or balanced 1/4" plug outs to hook up monitor speakers and a headphone jack for headphones.
A lot of audio interfaces work with a USB port (as long as you have enough USB ports and bandwidth) for any other USB devices, like a keyboard perhaps, or a mouse perhaps, anything that runs off of the USB port.
In a laptop you may only have one USB controller, while with a desktop computer you will usually have two USB controllers. If you have nothing else to run off of the USB port, then a USB audio/interface will probably work with a laptop computer, but again, you have the milli-amps and bandwidth usage of a USB port that is limited. All I know is my desktop computer which is standard - 500 milliamps - bandwidth handles up to 100 channels of music according to RME. That would be 50 in (channels - separate tracks) and 50 out (channels - separate tracks - but usually only two are used to go to stereo speakers being mixed down on the output, although an audio/interface may have more outs like 2 analog outs, headphones out, ADAT outs, SPDIF outs, depending on which audio/interface you get.
My unit (RME Babyface) has 10 ins and 12 outs (if I remember correctly)- 2 Analogs in (microphone, or synth or pre-amp or mixer in) and 8 ADAT in (depending on sample rate of recording which at higher sample rates - say 96kHz instead of 48kHz drops down to only 4 ADATs) while the same for outputs plus the 2 headphones out - 10 in - 12 out.
A USB audio/interface runs everything in and out of the computer through a USB connection on the computer for the music. Then there are other types of audio/interface using the other type connections like PCI-e or PCI (old now) or Firewire connectings if having a Firewire port in the first place, and most computers except maybe for a Mac do not have a Firewire port anymore, except to add a PCI-e Firewire card to the computer, but on a laptop that is not possible.
A sound card can be used, but be sure you know that soundcards are soundcards, while audio/interfaces usually most are professional type soundcards with the proper connections to plug in musical or microphone equipment (including having 48Volts for pro condenser microphones (not 9V like a soundcard for a computer mic) and outputs to go either to a stereo amp or studio monitor speakers.
There are many audio/interfaces made and a lot of people only need two inputs (at one time unless doing live band recordings where more may be needed) which is like L & R inputs for stereo signals of instruments but also can be used for mono-input sources of sound also. (not made for turntables which require a different pre-amp for old vinyl records).
http://www.sweetwater.com/shop/computer-audio/audio_interfaces/ Thunderbolt is a new connection method, but unless the computer has a Thunderbolt connector, it will have to be connected somehow to the computer, and although faster than USB, it is not really needed for recording music or sounds or whatever is recorded into a computer.
USB connections are widely used nowadays, because they are external to the computer and can be plugged in or not to the computer, but again laptops may only have one USB controller, while a desktop computer probably will have two USB controllers, which means more room for additional USB connections like more USB ports on the computer - mouse, keyboard - and then USB audio/interface means probably need two USB controllers because of milliamp usage and bandwidth usage on a computer. If you had a HD camcorder or camera that runs off of the USB port to put the movies taken or pictures taken on the computer - they take a lot of bandwidth and nothing else for that USB controller would be running if plugged into a USB port that also had the camcorder or camera plugged into the computer at the same time. In the end, all USB devices would not run or work because of that camcorder or camera taking all the bandwidth of the USB port at the time it is plugged into the computer downloading whatever into the computer. After unplugging the camcorder or camera from the computer's USB port then the other USB devices a person may have (even a printer nowadays) would work again, but not at the time the camcorder or camera was plugged in, or perhaps a computer printer would be the same way. Thus 2 USB controllers so the other devices like mouse and keyboard will remain working while the extensive bandwidth USB devices are being used on the computer at the same time.
So a mixer may work if it can be plugged into a computer and has sample rates like 44.1kHz or maybe 48kHz to probably a USB port for an interface, but an audio/interface like a USB device is actually made have the connections used for such purposes. It does not mean that a mixer will not work, some people use mixers, but then you will have to see what is better for you to use and how you plan on using any Mixer or Audio/Interface with your computer (or soundcard which I would not recommend for music but also have been used before with the mini-plugs (3.5mm connections instead of pro musical connections).