The difference between using で and と　is the difference between "we went out as a family" and "I went out with my family", I think of it as rather like the French "en famille". 家族で is very common expression and you can also hear this grammatical use when people refer to doing things as a group (グループで／皆でやりましょう).

It is another variation on the use of the で　particle to indicate the means/use of something to do something else (映画をテレビで見た、日本語で話した etc) although I must admit I find it easier to remember and understand intuitively rather than by trying to rationalise this  definition.  

(The easiest way I think of to explain the double use で is to compare it to the use of "how" in English: The question "How did you go to the races?" can grammatically be answered in two ways: "As a group/as a family" or "By car". The difference is that where as in English we use the same word to make two different inquiries, in Japanese we use the same particle to two different inquiries.)