The widow spiders, such as the infamous American Black Widow, have neurotoxin venom. The venom can cause muscles to cramp for long periods of time and may include the diaphragm. This will cause rapid, shallow breathing and can lead to death. The Brown Recluse and Hobo Spiders have necrotic venom. This means that the venom affects the cellular structure around the bite, but it can spread further into the body. The bite itself will develop a cyst with a white ring around it, giving it a characteristic "bulls-eye” appearance.
Most other spiders will leave a red welt on the skin that may itch. Generally symptoms will start to show from 30 minutes to six hours after the bite. An ice pack can help reduce the spread of venom. Also keep an eye on the bite itself, as it can become infected and cause a different rash of problems. It is possible, however, for your dog to have an allergic reaction so if the symptoms worsen take your dog to a veterinarian.
If the dog is behaving unusually, you can also contact an Emergency Animal Clinic. They generally will not charge for phone calls and can give advice including the administration of small doses of Benadryl.