I know that they are not the same, from personal experience.
More than once, when writing a mathematical proof, I developed the core part of the proof visually in my head.
In one case I remember, I did translate the mathematical language (which wasn't a verbal language, but a symbolic one that could be read) into visuals, worked out what the visuals would "look like" if the proof was invalid, then took the visuals and generated a mathematical description of what I was visualizing.
The process of manipulating that visualization was thought. At one point it was converted from a (non-verbal) language, it was manipulated as visualization, then it was then converted back to that non-verbal language.
I tried to explain the visualization steps -- the way I worked out how to do the proof -- to someone in order to explain how I generated the proof. I couldn't explain the visualization steps sufficiently in English, nor in Mathematics, despite the fact both her and I where fluent in both languages. When describing it, I used various and sundry terms to describe the things I was thinking about, but nothing fit; I suspect that is why communication failed.
The proof generated was clear, the problem statement was clear (both written in Mathematics), but the process used to generate the proof was not in any "language" either of us spoke.
Now, I guess you could stretch the definition of language to include visualizing things and motion of said things. If you define language to be "that which you use to think", that would make your proposal true.
(For the curious, the case I'm thinking about had to do with the existence of certain categories of continuous inverses of a particular kind of function. The visualization was imagining coloring things with a continuously changing tint and examining what it meant to the colors of the inverse. The proof had to do with how the category of functions limited the properties of the inverse, and hence how the inverse would paint the domain. Nothing really deep.)
Odds are you can experience this yourself. Think about doing something physical with your body, how it will move, etc. Try not to "narrate" it. Are you using language to describe how you imagine your body moving?