i cant speak of the 98's cause i own a 95 but neons are neons 95-99. i havent had much trouble with mine.
blew a timing belt which is bad cause the neon is an interference engine so i had to replace a couple valves on the head, no biggy, i blew one trans doin too long of burnouts (the diff pin slid out and took the diff out as a result). my ecu fried, had to get a new one, suspention is in great shape still, interior good, i am starting to get a leak out of the front seal on the cracnk. wheel hop is rough. they are an easy ride to woprk on really basic, parts are cheap to replace, and once you get the right headgasket, which dodge carries a great replacement, you really only have to worry about the timing belt being replaced every 60k miles or so.
i drag race mine any chance i get and i drive like i am out at the track all the time too, my car has it bad when it comes to me drivin but it still hangs.
wouldnt change it for any other sedan.

i would like to add that when i got the car a little over a year ago, it had 138k miles on it and now i have 148k.

 

In all honesty, no matter what you buy, it's gonna have it's share of problems. Sure the Neon has some common issues...headgaskets,sunroof problems, etc.
as for the transmission issue, look at the milage they are failing at generally 100k +, not too bad for a disposable car. They were never meant to be million mile cars, they were designed to be cheap basic transportation period. Neon, Camaro, Mustang, Porsche, if you beat it it's gonna break. So if it's a neon you desire, go for it.

 

I dont have a 98 Neon but i do have a 95 Neon and as Dazz said they are basicly all the same and with my 95 it has over 232,000 miles on it and its hardly ever seen the shop. only problem i've had with it is the speedometer and the radio going out. besides that its been the perfect car for me.

 

I have a 98 and Ive never had any problems with the tranny, the only problems I have had was from something that had to do with my heating and cooling system which where a new water pump and cracked head, but that came from driving it months on a blown headgasket that finally decided to crap out and even with the cracked head they where small cracks and a electrical problem which came to be a bad fuseable link.

 

i have 2 98 r/t's and just picked up a 99 sport 3 days ago for my wife....one of my r/t's has had the **** beat outa it sice the day i bought it....it only needed 1 new motor and 3 trannys.....but they were all my fault anyway, they werent from any mechanical defects...just very very hard driving...my other r/t gets driven alot better but still not that great and is a work car..well now anyways..but it hasnt had a single mechanical problem yet(124,000) and still runs like new, and drives decent..besides the back corner of a semi trailer that met the front passenger side at 95 m.p.h.....the 99 needs a new motor due tue a very bad rod knock from the person before me......but all these are self inflicted problems...it all depends on how you take care of your car, or how the previous owner took care of it

 

i have a 98 neon r/t the only thing i have had to do to mine was replace egr and i have 109*** but my friend has same car but year newer and he just replaced his clucth at 22**** somthing and he burns his tires every time he takes off

 

my 98 is in my garage getting a rear main seal at 63K. the sun roof crapped out 3 years ago, head gasket @30k..i do not run this car hard. it is relatively flat here, origional brakes still, but may throw rotors and pads on it just because it will be all apart pulling the trans out for the seal job.