There are plenty of different hand soaps out there, as well as hand sanitizers. Is there an advantage to soaps that claim that they're antibacterial vs soaps that just say soap?
In particular I'm looking at Softsoap who offers normal soap and antibacterial soap.
Examples:
Normal:
Antibacterial:
Short answer: There is no benefit for their use in households.
Long answer: These soaps (see here for the complete list) contain the so called quaternary ammonium compounds Benzalkonium chloride and Cetrimonium chloride which indeed have antimicrobial properties.
While they do not promote resistance to these compounds (see reference 1), their use is still not recommended, as their permanent use might dry out the skin, can cause contact allergies and the products released into the environment are also problematic.
There are two studies, which compared the use of normal soap (which has some antibacterial properties on its own) to antibacterial soaps in household environments and found no differences. See article linked in reference 2 for a summary and references 3 and 4 for details on the studies.
This doesn't mean that antibacterial soaps are useless at all, they simply make no sense in households. For hospitals or doctors they are an important tool to protect their patients before operations. But here detailed instructions for how long and how the hands have to be washed are provided. Hand washing is also followed by another disinfection step, which helps with efficiency. But this is very much different from the way people wash their hands at home. See reference 5 for some opinions here.
References:
Chris has correctly identified the antibacterial agent in the hand soap depicted in the image in the question, and therefore his answer is superior as a direct answer.
Nevertheless, other members of the Softsoap series of hand soap uses triclosan, 0.15% as their antibacterial agent, as seen in an image of their ingredient list on the reverse of the bottle.
(click for enlarged image)
Triclosan has been shown to be effective by Webster et al (1992) in reducing the incidence of MRSA infection as well as being milder to the skin as compared to chlorhexidine gluconate in a neonatal ward.
The average number of new cases of MRSA per week was reduced from 3·4 to 0·14 (P < 0·0001) in the experimental ward whilst no significant changes occurred in the control ward.
Triclosan has also been shown by Jones et al (2000) to be effective on a large variety of different bacterial, viral and fungal strains.
Therefore, the usage of triclosan is effective as an antibacterial agent, and would likely result in less bacteria on the hands after washing.
You may be interested to look into the Old friends hypothesis, since this is related to how the human immune system may respond to reduced biodiversity in the microbiota. The basic idea is that the human immune system is developed by exposture to the microbiota, and without exposture to organisms from our evolutionary past, immune system regulation might fail. This can then lead to poorly regulated inflammatory responses and maybe allergies or other chronic inflammatory diseases. Since the exposture to our evolutionary Old friends is lower in developed countries and urban areas, this hypothesis might potentially (partially) explain the increased prevalence of allergies in developed countries. Even though it is somewhat tangential to your question, and does not directly answer the question of the pros and cons of antibacterial soap, it should still be relevant for a more general case.
Two good starting points are Rook (2013) and Hanski et al (2012). Rook (2013) is a general overview of the theory with many good references to empirical studies that relate to the hypothesis, by one of the main proponents of the theory. Hanski et al (2012) is an interesting epidemiological paper, where they relate the allergic disposition of individuals to both the general biodiversity in their larger living environment as well as the generic diversity of their skin microbiota. In the paper, they find a positive relationship in healthy individuals between immune responses and the abundance of the gammaproteobacterial genus Acinetobacter on the skin, but they do not find the same pattern in atopic individuals. The study also finds a negative relationship between the environmental biodiversity close to the homes of study subjects and the frequency of atopy (so high biodiversity -> lower levels of atopy).
To be clear, this is still a hypothesis that needs further testing and support. However, there are many interesting case studies that can lend support to it, and indicate plausible mechanisms.
References:
Probably, it is worth to add some historic (and ironic) "perspective" to this question. It will probably explain that even without scientific research the answer to this question is probably "no" and that real use of bacterial soap should probably be reduced to hospital settings and not expanded to households as suggest promotions and commercials.
The historic perspective I wanted to add relates to the book "The Hermit in the Garden" by Gordon Campbell. Here live hermits are described (there were non-alive as well, this is why I call these "live"). Campbell cites another book - "Sir William Gell's A Tour in the Lakes Made in 1797":
"the hermit is never to leave the place, or hold conversation with anyone for seven years during which he is neither to wash himself or cleanse himself in any way whatever, but is to let his hair and nails both on hands and feet, grow as long as nature will permit them."
From this "work" description we can speculate that "dangerous" infections were not expected during such a long period of soapless live. This is why we could assume that any soap is probably of limited "antibacterial" value for otherwise healthy person.
(Remark: This "exclamation" not trying to answer the question in a scientific way, but just to add some "color" to the question and the answers posted. I hope, it is worth to place it here instead of a comment).
Human immune defense response is primed by cell damage products as it does not have prior knowledge about bacterial or viral "malware".
The whole purpose of disinfectant substances is damaging cells and potential organic disease carriers. As those substances are not discriminating, frequent use of them will prime your immune defense against benign organic matter as well. In combination with solvents that remove protective layers of fat and thus increase the permeation of this kind of cell waste into top skin layers, this sounds like a recipe for allergies and neurodermitic reactions in the long run.
There is a high correlation of countries (and eras) with high standards of living including household hygiene and autoimmune response problems. Finding actual causations is work in progress.
While that is still the case, it is my gut feeling that contact with disinfectants should be reserved to cases of tangible risk.
