It is a very interesting question. Both are Japanese forms of Martial Arts, or what classical Martial Artists like to call Gendai Budo. They are primarily a form of training to develop the moral a physical character of the practitioner.
Judo means "the gentle way or path" and was developed from Jujutsu. The adaptations made from Jujutsu involved developing a syllabus of techniques that were less violent or dangerous. Dr. Jigoro Kano developed Judo as a form of Physical Education and emphasized that his art would be able to teach the Japanese people about the Martial traditions and code of ethics of the Samurai whilst developing their minds and bodies.
In Judo, the practitioners wear a uniform called a gi (pronounced ghee). The syllabus includes throwing and tripping techniques (nage waza), locking and immobilization techniques (ne waza and katame waza) as well as two-man forms (kata) and breakfalls (ukemi waza). Judo has developed into a competitive sport which is 'played' in the Olympic games. Judo does have weapons in its syllabus but generally teaches them only to the point of learning to defend against them.
Aikido means "the way of unified energy or spirit" and was also developed from Jujutsu. The founder of Aikido, Morihei Ueshiba, gradually developed more circular and softer principles of the Daito Ryu Jujutsu system he had mastered p eventually adopting the name Aikido for his art.
In Aikido, the practitioners also wear a gi but most also wear the hakama which is the large pants worn by Samurai (as seen in many Samurai movies). The syllabus has throwing techniques nage waza) and locking techniques (katame waza) as well as the study of weaponry (bukijutsu) most usually the tanto (knife), the jo (5-foot stick) and the boken (wooden sword). Aikido focuses more on energy and how to use both yours and your opponents energy to either defend against or neutralize an attack.
Both systems are based on similar principles and the founders of each system conversed with each other during the infancy of each style.
Judo focuses more on the scientific principles of balance and co-ordination whereas Aikido focuses on harmonizing with the opponents energy.
Modern-day Judo is centered around competition whereas Aikido (most styles) do not engage in competition choosing rather to focus on training and personal development.
If you are considering which style to train in it would depend on what you want out of Martial Arts as well as what is available in your area. As a Martial Artists of some experience in both systems, I would recommend that you find (if possible) a traditional Judo school which focuses on training and development and then find an Aikido school which focuses on the same. After witnessing and then experiencing each style it will not take long to discover which one is right for you.