Often Karate is considered quite different than Tai Chi. However, the differences are not necessarily as pronounced as many think. First of all, it should be pointed out that the full name of Tai Chi is Tai Chi Chuan. Chuan means fist, or fighting style. Clearly Tai Chi evolved from "hard" martial arts and the early practice included "hard" training.
It is also interesting to consider that "Tai Chi" does not mean "soft" and definitely does not mean "yin". In fact, Tai Chi means "Supreme Ultimate", or yin and yang. Likewise, "Goju" means "hard-soft" (Goju Ryu is the karate style I study under Shihan Roger Wehrhahn).
Any style of martial arts, whether hard or soft, will contribute to health problems when imbalanced (too hard or soft). While I agree that karate can be considered a hard style, if the hard is developed while neglecting the soft, it can shorten one's life and contribute to health problems. If tai chi is practices without development of the hard or external practice the circulation will not be properly invigorated. It is a common claim that tai chi benefits the circulatory system, and that it accomplishes this largely throuhg training relaxation in general and during movements and postures. However, if an individual's constitution is already more relaxed (cold) than they might very well benefit more through the practice of karate, which invigorates the blood.
In my opinion the general health benefit of hard styles in the develoment of strength and the invigoration of the blood and chi; while the health benefit of soft styles relates to relaxation of the tendons, connective tissue, circulatory system, and mind and with the sinking of the chi. (Both hard and soft styles develop balance, coordination, and reflex.)
If one were to practice karate excessively without understanding the soft aspects and allowing the body to cool down, imbalance woudl result. Likewise, if one were to only practice tai chi (as presented to the modern public), they may lack the invigoration of harder styles of martial arts, hard work, and exersize like walking. Both hard and soft styles have hard and soft aspects. And as with all things balance is key.