Always a good thing to have a https for a website. Google tends to be more favourable towards https websites

It can have an impact. A huge one.

Links pointing to the HTTP version do not get credited to the HTTPS version and vice versa.

As we all know that HTTPS offers security, so it is definitely the choice to put you in Google's good graces. There are also some additional SEO benefits for you to consider.
1. HTTPS can increased ranking.
2. When traffic passes to an HTTPS site, the secure referral information is preserved.
3. It verifies that the website is the one the server it is supposed to be talking to.
Http is most commonly used to access html pages, and it is important to consider that other resources can be utilized through accessing http.

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This is not good for you because if your website is open both http or https version then Google behaving two different website and that will show duplicate content issue. So it's better to redirect 301 https version.

That`s highly unlikely that you would see a ranking improvement due to switching to https, but you totally shouldn`t have both versions available, as this may be confusing for the search engines.

Note that the referral data is being dropped while going from https to http, so you may face some issues in case you take part in an affiliate program for instance, and have to send referral traffic to an http website.