An A3 page has twice the area of an A4 one, so you should perhaps expect a printer capable of these page sizes to cost around twice the price. In fact, A3 and A3+ printers tend to cost rather more than double their A4 equivalents, but part of that is down to economies of scale. There are far fewer people wanting to produce photos bigger than A4, so these models attract an extra premium.
In many cases, large format printers use the same basic mechanisms as their smaller, A4 counterparts, but with wider carriages and platens and sometimes extra feed trays to accommodate a wider range of paper sizes.
To round up, if you want to print batches of event photos for your family and friends, a dedicated, small-format photo printer is an ideal choice. If you want to print photos up to A4 but still have the facility to print general correspondence, school work and other plain paper documents, pick one of the photo models from the manufacturers' mainstream A4 range of printers or all-in-ones. If you need larger prints and if you're photographing in a semi-professional capacity, you will benefit from the flexibility and print quality of larger format printers.