There must be data out there about this. Do more people just hit enter after their search term or do they hit the "Search/Submit" button?
I'm not aware of any statistics, but I think Luke Wroblewski would point to the need for "a clear path to completion", which an explicit button provides. A submit button in a form (even a simple search form with a single input) does more than just provide a control for someone to press - it marks the end of the form, clarifies the purpose of the form (assuming the wording of the button is more descriptive than "submit") and serves as a call-to-action to submit, even if the user then decides to perform that action using their keyboard.
This is a fun question, and we ought to be able to answer it without disclosing proprietary information.
Here are some sketchy observations, though I hope others with access to their own analytics data can compare, critique, and share. I'm not confident in the reasoning here. Perhaps this will get us thinking.
After a glance at our search stats (from covenanteyes.com) over the past month, I see an indication that using the enter key is a more frequent action than clicking the button itself. Based on the 2884 searches I'm looking at, it seems that 752 were triggered by a click of a search button and 2132 were triggered by the enter key. So, about 26% used a button and 74% used the enter key.
What leads me to think I see this data? The search form in the top-right corner of our web page has the submit button as an image. As it is named "sa," I can track that in the query string of the search results page. As an INPUT element with a type of IMAGE, the form submits the coordinates of the actual click. So when I click the button, I see something like this in the query string: sa.x=7&sa.y=9. The x and y values vary based on where on the image the person clicked. When I hit the enter key, I see sa.x=0&sa.y=0.
It is possible, though I believe rare, that a person will click the very top left pixel of the button and also get x=0&y=0, but it is so unlikely that I've ignored it. People tend to click somewhere in the middle of the target.
Here is why I seriously question this data. We had 4194 total unique searches, but the URL patterns aren't consistent: they don't all have the sa.x and sa.y parameters. I only evaluated the 2884 that contained the x and y values. Now, when I simply tried the searches to see what happened to the URLs, the x and y values did in fact appear. They always will when those forms are used, because that's how form submissions work.
But still, pretty big assumptions here. Are there other times when they will appear which can also cause the zero values? That would throw off the proportions, perhaps in a large way. Anyone want to grab the baton and run the next leg of reasoning this through?
What usage patterns have I not realized? Why are those parameters missing in the other searches?
Can someone else look at their own search data and report back?
(Aside: no doubt Google's live search dramatically changes this, as every keystroke seems to refresh the search results. I'll bet button use in that context plummets.)
It would be interesting, especially to measure it against other things such as demographics. I suspect the results would be well correlated with demographics, which might mean you could make a reasonable guess if you knew your audience.
@BryanRobinson's comment on @MattObee's answer matches my experience having worked on designing systems for older people with little (or no) computer experience that novice users click, experienced users hit enter, hence my expectation that demographics will have a large impact.
As an aside, StackExchange's design without a button, just a field could be taken as a (probably quite accurate for this community) expectation that SE users are experienced users. It may be a bit less accurate for som other SE communities though, such as english.stackexchange.com.