As you would like to record vocals (nothing more) then it's pretty much a coin flip between the 2 machines. Pick one which is less expensive and that would be the 414.
Note that neither has inserts for your compressor so you'd just run it direct mic - preamp - compressor to the XLR mic input. You'd then, when you have finished recording your vocal, use either the L or R RCA outs to go to your computer's soundcard. Really, the only feauture you'd find beneficial on the 424mkIII is the automatic punch in and punch out with rehearsal. That is if you wish to do a lot of "inserts" to make the vocal "flawless".
Now, what I just thought of is that the 424mkIII also has 4 RCA tape outs that you can run into 4 RCA's of your soundcard (if you have that option). That could be beneficial too. But I'm kinda confused .... Do you wish only to do your vocals on one of the tracks or would you take up all 4 ? Sorry to be writing in a chaotic type of manner but I'm just writing as things "come to me". Another thing you may try on the 414mkII is record your vocal on 4 tracks and then dump them to one track (via the L or R RCA outs) but this time ride the faders to come up with one "perfect" take taken from all four. This may take some practice but it's an old recording school type of trick that works good. That way you may not need to spend more on the 424's automatic punch in feature. Seriously, I think you could do pretty much similar things with both machines. Again, it's a coin flip .... Tho I wonder why you don't want to pick up one of those stereo consumer open reel decks ... Can be found locally too and if in good condition will suffice for what you wanna use it for but will sound much better than teh 424 or 414. They run at 7 1/2 ips and have 2x the track width and are arguably much easier to use than the cassette portas for your purposes.
With regards to your last question: Usually open reel tape means that the machine that it is being used with is operating at a much faster speed (tape speed) and/or has wider tracks than a cassette (recorder). This means more space is being given to frequencies and they become more and more defined and truer to life as the speed and track width increase. Generally cassette is Low Fi and open reel is Hi - Fi. In your example, the
Tascam 388 open reel will sound more Hi-Fi than your cassette porta because it dedicates TWICE the amount of tape to the source sound but even more so, electronics of the 388 are HUGELY superior to the 414 or 424 making the sound even better.
~Daniel
Will you sing to music ? Will there be any other tracks on the tape machine you wish to use or just bare vocals ? Please explain ...