Technology Marches On: If mobile phones were as common back when this movie was made as they are today, this would have been a very short movie.
And funny enough, there is a cellular phone ad at the L-Train stop at the end of the film. Also worth noting Due Date borrows quite heavily from this film, and that's from 2010.
Neal may have been able to contact his wife but he would still have had the inability to get home in time for Thanksgiving, which was his biggest worry throughout the film. A phone wouldn't have made the planes run on time or stopped his car from catching fire.
… but it would have made some of the troubles associated with those situations resolve much more quickly—for example: on finding that the car he rented didn't exist, Neal could have called the rental company for a pick up.
Finding the next flight or the nearest hotel? There's an app for that.
Troubled Production: The experience of making the film was not a happy one for John Hughes as many shooting days would either be lost or delayed due to weather issues or having to work around certain loopholes (for example, a sequence involving a train had change shooting locations due to a lack of snow and the crew had to create a train route from scratch as the local train company wouldn't allow them to use theirs). Also, the rough cut ran over three hours and the film spent many months in post-production so to cut the film to a manageable length (this is also why references to Hughes's next film appear, as it has begun production right after this film finished filming). In addition to these problems, Hughes was also smarting over the fact that his long term business relationship with Molly Ringwald had gone sour after she turned down the Lea Thompson role of Amanda Jones in Some Kind of Wonderful. Hughes was so upset over the rejection that he never worked with Ringwald again for the rest of his life.