It's been implied in other answers, but not stated explicitly, that  "every time" and "all the time" use two different meanings of the word "time". As Barmar said in a comment, the former is countable (like beans) and the latter is uncountable (like water).

The construction "every X" forces us to interpret X as countable. Specifically, "every time" uses meaning 
3.4 below (abridged from the ["time" wiktionary entry][1], with emphasis added) whereas "all the time" uses meaning 1 (or possibly a version of 2).

[1]: https://en.wiktionary.org/w/index.php?title=time&oldid=65600074

> *time* (countable and uncountable, plural *times*)
> 
> 1. **(uncountable) The inevitable progression into the future with the passing of present and past events.**
> 
>    * ***Time** stops for nobody.*
>    * *the ebb and flow of **time***
> 
> 2. A duration of time.
> 
> 
> 3. An instant of time.
> 
>     3.1. (uncountable) How much of a day has passed; the moment, as indicated by a clock or similar device.
> 
>     3.2. (countable) A particular moment or hour; the appropriate moment or hour for something (especially with prepositional phrase or imperfect subjunctive).
> 
>     3.3. (countable) A numerical indication of a particular moment.
> 
>     3.4. **(countable) An instance or occurrence:**
>          
>       * *When was the last **time** we went out? I don’t remember.*
>       * *see you another **time***
>       * *that’s three **times** he’s made the same mistake*
>       * *Okay, but this is the last **time**. No more after that!*
>
>    3.5 (UK, in public houses) Closing time.
>
>    3.6 The hour of childbirth.
> 
>    3.7 (as *someone's time*) The end of someone's life, conceived by the speaker as having been predestined.
> 
> 4. (countable) The measurement under some system of region of day or moment.
> 
> 5. (countable) Ratio of comparison.
> 
> 6. (grammar, dated) Tense.
>
> 7. (music) The measured duration of sounds; measure; tempo; rate of movement; rhythmical division.

It's probably worth mentioning that you *can* use "every X" to get the sense of "all the time" if you divide time into countable units. A person who has (and is hopefully seeking help for) Generalized Anxiety Disorder might say:

* *How often do I feel anxious?  **Every second** of the day!*
* *How often do I feel anxious?  **Every** waking **moment**!*

If you want to keep the word "time", you need a construction that does not force a countable interpretation, such as:

* *How often do I feel anxious?  **All the time**!*