I'm no expert on Short Hills/Millburn, but I can speak on behalf of Scarsdale.
1) Metro North is the most reliable of all methods of mass transit amongst the outside of NYC Suburbs (LIRR, NJ Transit Train, NJ Transit Bus). If you take a train from the Scarsdale Station it will be as fast as 30 minutes to Grand Central (one express stop that may stop at 125th St & then Grand Central; some don't even stop at 125th St) or as long as 50-55 minutes if it makes every local stop. Trains are sometimes delayed but unless there is a lot of snow, it's rarely longer than 5-10 minutes.
It shouldn't be a deal breaker, but if you work near Grand Central, it makes more sense to live in Scarsdale. If you work near Penn Station or near one of the Path stations, it makes more sense to live in Short Hills/Millburn.
2) They have similar demographics. High Jewish & Asian populations. Short Hills/Millburn has a slightly higher Asian population than Scarsdale (I believe it's ~ 33% in SH compared to 20% in Scarsdale). This is due to there being more Asians in NJ than Westchester.
Below is the NYS Education data for Scarsdale (includes demographics):
https://data.nysed.gov/profile.php?instid=800000034921
3) School districts are both roughly the same size (~4500 students in SH & 4750 in Scarsdale).
4) Houses seem to be slightly cheaper in Short Hills/Milburn than Scarsdale. Property taxes are also cheaper (~2% in SH vs ~2.4-2.5% in Scarsdale).
5) I'm sure the housing market is extremely competitive in SH but in Scarsdale, inventory is at all time lows. Houses under $2mm have 10+ bids before the weekend is over. Even the $2-$3mm range is extremely competitive and gets into bidding wars.
6) I can't speak on SH but Scarsdale leans much more Democratic than Republican.
If you have more specific questions, feel free to ask here or PM me.