I can't give you a definite answer, I can only speculate but I would guess that it would become slightly snappier if you're browsing a few simple pages as RAM is faster than HDD/SSD/permanent storage. However RAM is limited, compared to permanent storage, and once that is filled Opera, or any other browser for that matter, would no longer be able to hold the data, and that simply means that the data would have to be re-downloaded. So you'd have tabs that reload (only the images and stuff on the page, not the page itself I would guess) when you switch to them, which in turn is slower than HDD access.
So it depends on your surfing style, it could be faster, it could be slower with the disk cache disabled.
As a sida note: if the SSD is modern; latest generation or so, limited rewriting should not be a noticeable problem. It's mostly the older models that suffers from write-wear.
1723 day ago