The story would be very different. Harry would have a new set of values instilled in him by the royal family. This could affect his decisions regarding the wizarding world. For example, he might be more hesitant to engage in dangerous battles as he now has a royal position to uphold.
His living conditions would change drastically. Instead of living with the Dursleys, he'd be in a palace. He'd have servants and a much more luxurious life.
If Harry was adopted by the Queen, his social circle would expand to include important Muggle figures. This could lead to a new form of alliance between the wizarding and Muggle worlds. Also, his upbringing in the royal environment might give him more confidence and a different leadership style when it comes to fighting against Voldemort. He could use his royal connections to gain more support for the wizarding world from the Muggle side.