No, Taekwondo and Karate are different martial arts. They have a number of similarities, but also a number of key differences.
Taekwondo is genealogically related to Karate – many the people who developed Taekwondo in the 1950s and 1960s had trained in Karate in Japan. The basic structure of Taekwondo is the same as Karate – there are several stances which are the same, and basic hand techniques such as punches and knife-hand strikes are the same. Taekwondo clothing is also similar to Karate clothing, and Taekwondo practitioners also wear coloured belts to indicate level of ability, among other similarities.
But there are a number of key differences between Taekwondo and Karate. The most visually striking is that Taekwondo places far more emphasis on kicking techniques than Karate, and less emphasis on hand techniques. Taekwondo has a greater variety of kicking techniques. This is very apparent in sparring – in World Taekwondo sparring competitions, kicking techniques are used far more than hand techniques.
Taekwondo also uses an entirely different set of forms. Forms are predetermined sequences of movements used for training and competition. And since Taekwondo is from Korea, Taekwondo uses Korean language terms to describe movements, and follows several Korean cultural conventions.