A sequencer is a midi device. It records and plays back sezuences of midi messages. And also allows you to program the sequences. But you're not recording audio. Just midi messages.
Some come with an internal audio source, others are just sequencers and have to be connected (via midi) to a module or synth of some sort to allow you to hear something. So, you just program what notes have to be played where, and what parameters (CC) have to be set to which value at what time. The sounds are coming from your module or synth in this case.
Software sequencers are the software version of this. They will play back the sequences using a midi out port, or by sending some sort of virtual midi signal to a software synth or software sampler. Or they can use your soundcards builtin gm-midi sounds to play it back.
Alot of audio programs have a built in sequencers. Cakewalk allows both audio and midi to be recorded if I'm not mistaking. Some scaled down versions of programs allow only for the sequencer to be used, no audio.
An audio editor is basically a program that allow you to edit audio. So that would not include recording, certainly not on multiple channels. Just cutting, pasting, adding effects,.... Kindof like a photoshop for audiosamples. But most programs allow alot more I guess. I'm don't know that many programs.
I guess a softwarestudio would be something like cakewalk, that allows you to record multiple tracks, add effects, mix the tracks, maybe has a builtin sequencer and audio editor.
I'm not an expert on the field, but I assume that I'm pretty much correct here... Correct me if I'm wrong.