From a scientific standpoint both have a beneficial role to the human body.
The body does not synthesize carotenoids. These are obtained by us from foods like tomatoes, fruits, and vegetables or dietary supplements. Ketchup when made from tomatoes is rich in carotenoids.
Carotenoids are an essential component of the human diet. They act as antioxidants within the body protecting it against cell damage. Beta carotene gets stored in the liver and is converted into the body as needed into Vitamin A. It is a provitamin (i.e. a precursor to the vitamin). Vitamin A is converted into a light-sensitive pigment — rhodopsin- present in the eye retina.
The daily requirement of beta-carotene is just 1000mg for males and 800mg for females. Excess Vitamin A is harmful to the body, but carotene is non-toxic.
It is long considered that processed fruits and vegetables have lower nutritional value than fresh commodities due to the loss of vitamin C during processing. A study showed that thermal processing elevated total antioxidant activity and bioaccessible lycopene content in tomatoes. No significant changes in the total phenolics and total flavonoids content, although a loss of vitamin C was seen. This study helps clarify doubts in the minds of consumers concerned about the loss of nutritional value of processed tomato products (Dewanto et al., 2002).
Alpha-carotene too is a precursor to creating Vitamin A in the body but is far less common than beta carotene. Both alpha and beta-carotene are fat-soluble and are best consumed with some form of fat.
A teaspoon of B.alba (white mustard) seed contains 87.1 mg of omega-3 fatty acids, 84.2 mg of omega-6 fatty acids, 22.2 mg of potassium, 27.3 mg of phosphorous, 9.7 mg of magnesium, and 16.9 mg of calcium.
Scientific evidence in support of the medicinal properties of mustard is as yet inconclusive.
I would therefore conclude that when consumed in moderation both ketchup and mustard are nutritionally valuable.