Does board thickness matter when choosing laminate?

It does! If your new flooring is a different thickness than your old flooring it can change the way things look. Board thickness can affect the clearance of doors (you may have gaps if it is thinner, or door may scrape the floor if it is thicker) refrigerators may not fit under cabinets, dishwashers may be harder to remove, etc...

Thicker boards can also make baseboards appear shorter as more of the bottom has been covered. Heights from room to room will also be affected. 

This doesn't mean that you can't choose a flooring that is a different thickness than what you had before, it just means you need to prepare for it!