I think that one of the biggest differences is that Lexington, although not a "big city" by any means, is definitely a lot more of an urban downtown with outlying suburban areas. It has several large shopping areas, an extensive number of restaurants (both chains and local), and many more amenities. The university areas alone have large populations. Many people who live here have moved or transplanted from other areas, and there is a lot more diversity.
Although Georgetown has been experiencing a boom in the last 20 years due to the huge Toyota facility, is still has much more of a small town and rural feel. Many residents are native to the area and the area still has a strong farming and rural background. Although there has been growth in both retail and residential, (including a large retail area that is currently proposed), it doesn't offer as much - shopping opportunities are limited to Wal-mart, K-mart, Kohl's, Lowe's, Tractor Supply, Kroger (a grocery chain), etc. Restaurants are also limited - there are a few mom-and-pop style diners as well as a local coffeehouse, plus most typical fast food offerings. There is an Applebee's, a Ruby Tuesday, a Cracker Barrel, a Golden Corral, a couple of Mexican restaurants, some low-end Chinese and Chinese buffet places, etc.
The truth is that the towns are not that far apart that you can't go between them - from Georgetown you can usually be at Hamburg in Lexington (a big shopping district with many chain restaurants and lots of typical suburban big box stores) in 30 minutes or less. But the overall attitude, lifestyle and general tone of the areas are much different. Not that one is necessarily better than the other, but your family probably has a preference. Georgetown is not a typical exurb (like Oldham County is to Louisville) where basically most people consider themselves a part of Louisville but just prefer to live in a more pastoral setting. Georgetown is still a working farming community, and although many residents will frequently drive back and forth to Lexington, they do consider themselves a completely separate entity.