Mic particularly makes a difference, and although the general culture here on GS is "pick the right mic for the right voice" you're not always afforded that luxury, so generally speaking, throwing up a GOOD mic will give you the least hassle during mixdown...it doesn't have to be 100% the most amazing perfect match because the fact is there's so many variables that can dictate that, and where 1 shines in some instances another might shine in others, so it kind of becomes a taste issue.
That being said, I've found that when you throw up a shitty mic the artist more than anyone notices, so that may effect that performance.
There's a lot of other things that effect a good recording higher on the priority list than Preamp in my eyes, namely tone and performance of the artist, and a lot of that can change day to day as well.
For example, Joe's voice generally doesn't sound great the day after a performance (a lot of people have this problem) so you could throw up a billion dollar mic and a gazillion dollar preamp and still have a tone that doesn't sit pretty. Also, the way a rhyme is written can effect how something sounds, the placement of consonants and such, the GREATS are aware of which combinations of vocal sounds sound good in their voice and which don't.
Pitch is also important, another artist I work with a lot, OC (DITC) is very cognizant of his pitch, and well run through different takes for a while just getting the pitch right for the song...
I recorded some vocals of Joe's recently on his single with Lil Wayne with a C414 through stock C|24 preamps, which I would describe as a far from ideal signal chain, but it still mixed fine with the stuff I recorded on the Neumann through the Manley.
I find that the advantage of using a nice Preamp is that it generally has more flexibility, Low Pass Filters, Pads, Gain and Output controls all come in handy when you're trying to get the best signal possible. The difference between a Neve and say a API to me is greatly exaggerated and ultimately, in the context of a mix won't make a huge difference.
On No Love Lost (Joe's new album) I used the following preamps: Avalon 737 Amek 9098
Digitech VTP-1 API 3124
I dare someone to even begin to guess which was used on which song...
Compressors are very different though for sure, I like to compress during tracking, but again any capable compressor will do honestly.
So I guess to sum up, I'd say #1 is writing, #2 is performance/tone/feel/timing #3 is Engineer #4 is gear...
If I were someone who was a rapper looking to buy gear or something I'd generally recommend getting the best you can afford. If spending 10k on a U47
is nothing to you by all means buy the great mic, but I think for most people getting something light like say a 4040
or a Sm7
and an Mbox for cutting demos and figuring out things at home is more than enough, if you're gonna spend a grip of $ I think spending it with an engineer at a studio will get you further than spending a grip on shit you don't really know how to use and don't really have the time to learn, or cut the songs on your decent setup and give the files to a mix engineer to polish for you.
If you're an aspiring engineer buy the best stuff you can afford while keeping in mind gear may matter to your potential clients (depending on how good you are, or how much they trust your ability to give them a great product).
If your an aspiring producer id probably lump you in with the "aspiring rapper" category above.
Hope that makes sense and helps, if you have any other questions let me know