-Which one is more professional?
I'm not sure it makes a difference anymore. Screenprinted neck labels are common from the cheapest dodgy imports through to overpriced designer tees.
Consequently some customers associate it with high-end brands, others with low-end brands. Depends on the person.
-Which one would be cheaper?
Depending on who you get to do the labels, it could be either. As far as the cheapest price you'll find on either is concerned, screenprinting is probably cheaper. But if you contact any given random local contractor, it could be either.
Neither is expensive.
-What are the pros and cons of each method?
I'm old-fashioned, so I still associate a good woven tag with quality. Most young designers don't bother, or even know how to get things like that done. It shows they're more heavily invested in their brand.
Some customers hate woven tags because they irritate their skin.
Woven tags will often wear better in the wash, which can keep your product looking fresher for longer. (technically it could also have a legal benefit since the tag is supposed to last the useful life of the t-shirt; but that's never going to be relevant, even if it's technically true)
Mostly the pros and cons come down to perception though. Tags are an integral part of branding your product, and help you position your shirt in the market place.
For example, my design style tends to be simple (simplistic even), minimal, etc. So I like woven tags to convey an image of quality that furthers that idea that 'Sure, this stuff is basic... but that's because I want it that way, and I know what I'm doing. Trust me and go with it.'
(whether that's true or not is up to the customer, but that's what I'm going for)
But if I had a brand like Threadless, with big bold young graphics, I would probably print my neck labels, incorporating a graphic that riffs off the main shirt graphic (as they often do). I'd also include a hang-tag that either incorporated the complete art like a tiny-poster, or concentrated on an element I wanted to feature or re-contextualise.
I guess ultimately what I'm saying is if I was positioning the clothing as being about fashion, buying a piece of clothing, etc. I would use woven tags.
If I was positioning the clothing as being about the graphic design, and the t-shirt was ultimately just a convenient medium for you to buy and display that graphic design (it could just as easily have been a poster), then I would probably use printed tags.
There is no clear right and wrong on this one though - that's just how I approach it.