I thought the Delta 66/OMNI studio had no MIDI interface. Yup, just looked at the website, no MIDI.
When you say something can "handle MIDI," this is a bit vague. As James said, MIDI is a bunch of information about the music performance, not the audio of the performance. It's analogous to a player piano, where the piano roll has no sound in it, but it actuates the keys in a player piano and triggers the hammers to fall on the right notes at the right times. The piano roll is recorded information about the performance that allows it to be recreated on any compatible player piano.
In the same way, MIDI data allows a performance to be recreated a MIDI-compatible device. Computers use recording software to record the data from a MIDI instument coming into the MIDI In port, and once it's recorded it can be sent via the MIDI Out port to any device configured to receive the data.
Ever since the first SoundBlaster, consumer soundcards have had built-in MIDI synths in them, usually pretty cheesy ones that were more or less useless. A few cards came out in the 90's that had better MIDI playback potential -- the Turtle Beach Multisound and the SoundBlaster AWE series were notable examples.
As computers got faster, it became possible to run a "virtual" MIDI synth that was a software application, rather than a hardware device. These softsynths, as James noted, run in the computer, and you direct the recording software's MIDI data to it sort of through a virtual MIDI port withing the computer. In this case the actual audio output of the synth is piped through the WAV data stream just like recorded audio data would be. This kind of MIDI device can be run without an external synth or a MIDI out port, though you would still need a MIDI interface if you wanted to record your own keyboard performances as MIDI data.
With the Delta66/OMNI, Logic, and a fast enough computer, you can get excellent MIDI playback using a VSTi (VST instrument -- VST is the plug-in architecture that Logic supports). But there's no MIDI interface so you can't send it out to external devices or hook a controller keyboard up to it to get the MIDI data it spits out when you play it to the computer.
One more note: "MIDI controller" usually refers to a keyboard that has no sound on board - it is purely a generator of MIDI messages but does not create any sound in response to them, as James mentioned. Many MIDI synths have keyboards and sound-making capabilities, and they are perfectly good for using as a controller. Then there are the alternative controllers -- wind controllers, drum pads, guitars with pitch-to-MIDI converting pickups and electronics hooked to them...
It delivers MIDI data. Speak MIDI language. MIDI data devided into several kind of type, eg. Note on/off message, MIDI control change (CC), bank/program change, etc. etc... These days, MIDI controller are shiped in various type, and used on purpose. If you want to input MIDI notes into your sequencer, then you need "note input" kind of MIDI controller (eg. keyboard, drum pads, etc). If you want to control/sync playback and record, then you need MIDI controller which send those kind of data. Some MIDI controller can be used as "note input" device, control CC, start/stop and sync sequencer, etc. Your sequencer (PC) will recieve these data and respond (or record) to the message delivered.