Your score will definitely change especially if it is the oldest credit card in your profile. When you close old card, you shorten your credit history and lower your credit score. If it is a new account, then you can freely close it. Next thing to consider is your credit utilization ratio. Calculate what your credit utilization ratio will be after you close this card. If it is going to be higher than 30% then your credit score can be lowered significantly. Your credit score changes all the time, so if you want to close this card, just do it. You will simply need to work a bit harder to raise your score back.
In the future to avoid the situation when you don’t need the card for anything other than just a sign-up bonus, get a card that you can and will use for your everyday purchases. For example, get the Discover it® Miles. The rewards are 1.5 miles for every purchase, and in a way you also get a sign-up bonus – you build it yourself from your spendings as Discover simply matches all you’ve earned during the first year (provided you a new cardmember). There’s no limit to how many you can earn and the miles never expire – thus, you’ll never have to close the card because it’s reached its limits.