The two main points (against) within the UK can be taken from the Public Health England report that PETER01 linked to:

"Under the current regulatory system individual e-cigarette products vary
considerably in quality and specification. We also do not yet have data on their long-term safety. " Without regulation, we don't actually know what compounds new brands of e-cigarette are putting into the air. The 'best estimate by experts' is based on some early brands which have been tested, but the tests don't cover the whole market. There is very probably a lot lower risk than smoking, but that doesn't mean it should be allowed in public spaces.

"The potential of e-cigarettes to help improve public health depends on the extent to which they can act as a route out of smoking for the country’s eight
million tobacco users, without providing a route into smoking for children and non-smokers. Appropriate and proportionate regulation is essential if this goal is to be achieved. " In the UK, this is the key statement. The use of e-cigarettes is not going to be allowed in public places if it can be shown that it normalises the actions of smoking, because that will encourage young people to take it up. Even if banning also encourages young people to 'vape' purely because it is against the rules!