I've been to Disney during Christmas week (Dec. 24th to Jan 1st) the past 4 out of 5 years. Honestly, while it's a VERY busy and packed time, I would not trade that week in for any other vacation or time of the year.
First, a plug: I'll be releasing a book sometime this summer that will tell you everything you need to know about Disney during Christmas. I've decided to make it free. If I'm allowed, I'll post the link and everything once it's done.
Second... you will get mixed reviews on Disney during Christmas week. There are some that have never experienced it - have only heard about the sheer volume of people that flock to Disney every year during this time. These are the people that go off of hearsay and will give you the general "it's very packed based on [inset website's crowd calendar.] Then, there's the person who experienced it once and hated it. Maybe they tried it or had different expectations. Then, there's people like me who loved it..
Here's the thing - having experienced Disney so many times, you're already "cut out" for the trip. I will never tell someone NOT to go, but Disney during Christmas week is not for first timers. You will benefit from having seen and done everything multiple times already. Many people go for their first time or go and expect to see and do everything in the park. Then, they are disappointed when the crazy wait times keep them from going on Peter Pan, or riding Space Mountain. Well, if you plan your day right, you can see and experience more than you do not; but, having experienced everything already gives you the ability to see the 100-minute wait for Peter Pan and say "well, we've been on it so many times, we can skip this time!"
I won't sugar coat it - it gets PACKED. But staying onsite gives you the upper edge. My family gets so much done during EMH. We go to the park early in the morning, stay until 12 or 1pm and then head back to our room to to relax and maybe catch a nap. Around 5 or 6 pm (depending on our dinner reservation) we head out to dinner, and then out to whichever park has EMH. Disney expands EMH and has a lot of late nights. The Friday after Christmas is almost always open until 2 or 3am for EMH.
It will be crowded on December 21st, 22nd and 23rd, but the real crowds will not roll in until December 24th. The only downfall is that the 21st falls on a Sunday - more people are bound to start their vacation on a Sunday or Monday than a Tuesday.
You'll see a slight lull after Christmas (comparatively speaking.) The 26th and 27th won't be too bad, but the crowds will kick in again on the 29th, 30th and 31st. You would be leaving before the storm. NYE is actually more crowded than Christmas Eve and Christmas day. I think Christmas day is a majority of tourists and people from out of the area. I think NYE is a combination of locals AND tourists. As a result, the parks are CRAZY crowded.
As for driving... well... My family drives from New York. Honestly, I loved it. Some of my best memories as a child was the drive down to Disney World. But, having flown down the past few trips - it's SO much easier. Board the flight and within 2-3 hours, you're there. Driving would add/take away a day or two from your vacation! Spring for the flight - it's expensive, but ever my mom who hates flying would NEVER look back!
As far as it feeling like Christmas - Imagine the excitement when you're younger of running downstairs (or into the other room) and seeing all the presents under the tree for the first time. Cinnamon rolls are baking fresh in the oven and the smell overwhelms the house. Well, it's a little different. Christmas morning you'll wake up in your hotel room. You'll board the bus. Everyone is so happy. Strangers are wishing each other "Merry Christmas." You arrive at Magic Kingdom. You enter the park to jovial Christmas music playing. This time, instead of seeing the tree for the first time, you see the castle. The smell from the confectionary and homemade waffle cones from The Plaza fills Main Street. Suddenly, you wonder how you could have possibly spent Christmas at home all these years. The potential for 70 degrees and sunny doesn't hurt, either!
P.S - there's so much more to do in Disney during Christmas week than the parks! With an older family (younger than mine, but older than the normal family) you can enjoy the resorts and their decorations, and enjoy the nice days. I can't promise nice weather. LAst year we were blessed with gorgeous, warm weather all week. Next year it may be colder than Colorado all week - but if you get some temps in the 70s, enjoy it! I loved leaving the parks last year and heading back to the pool. We never went in, but I enjoyed hanging out and even going for a jog.