Yeah, I've also noticed that. My guess is that it's because, when you Google something directly from your browser, Google is using your personal preferences (specially if you are logged into your Google account) and user metadata. However, if you do it via the API, the custom search engine you created is what's controlling the search.
If you don't want those Google+ results to appear when using GoogleCustomSearch from the Wolfram Language, you can do:
gcs = ServiceConnect["GoogleCustomSearch"]
gcs["Search",{"Query"-> "Your name","Site"-> Except["plus.google.com"]}]
Feel free to ask anything if you still have issues with this or other services. Note that for web search, BingSeach is also available