A question for (traditional) Fighting Game Enthusiasts (console preferrably)

[L.C.E.C.]

Hi, I like fighting games. I'm not a "pro" or anything, and I don't play tourneys... I love SSBB and still play that actively, but this thread is not about SSBB and SSBB-esque games, it is about more traditional fighting games...

I used to own people at all the local arcades as a pre-teen to teenager (10-14). Mainly I liked the Street Fighter and Marvel vs. Capcom series. I was decent at Tekken and played Mortal Kombat a lot just for fun... But when it came to Street Fighter II or MvC II, I'd beat 25-year olds all the time... I also played SFO (Street Fighter Online) for a good few years. I got a GameCube for the first time (I had played on N64s and PS1s at friends house at age 13 and expanded my tastes in terms of game genres (including the more "casual" fighting games, such as SSB, DBZ Games, and Naruto), fell in love with SSBM and tons of other games... 

I recently bought SSIV Arcade Edition (a few months ago) for PS3, and it's pretty good. I'm starting to get back into it... I'm still kinda rusty (I get pwnd online about half the time), but I'm not terrible. Anyway, I recently bought my Dad Namco Museum Megamix for Wii, and my Mom (she plays PacMan a lot) complained that it wasn't quite the same as using an arcade joystick... So, I did a little research and found this baby at GameStop for $13:

http://www.amazon.com/Wii-Tatsunoko-Capcom-Arcade-FightStick-Nintendo/dp/B0032ISZL8

There were two within my state (about 50 miles away), and I had my gamstop ship them over to my local store and bought both of them. I converted one to a 4-way (for games like galaga, pac-man, dig dug, etc) by opening up the inside and turning the restrictor plate. I tried out the other one for a few weeks before I converted it to a 4-way also, so that if two people were playing, that they wouldn't have to unplug their remote so the other person could use the fight stick...

So, in my using the fight stick, I didn't really see the big deal that most fighting game enthusiasts make. I understand the idea of digital input vs. analog (which is why I like the D-pad for pulling off hadoukens), but I don't understand why it's "so much better" than a standard controller (except xbox, i understand because the D-pad is terrible)... 8 buttons, a digital input analog... There are 4 face buttons and 4 trigger buttons on a Dual-shock, and a digital D-pad... Same number of inputs (and 2 additional analog sticks)... What's the big deal/advantage? Maybe I'm just used to console controllers now instead of arcades, but I just don't get what the big deal is... Will someone explain it to me?