user4661: I want to run the 5V board at 3.3V because I can get the 5V version cheaper than the 3.3V version.
I read that you can power the Arduino Pro Mini with a battery by connecting the battery to the 5V pin, bypassing the regulator.
Somewhere else I read that the only difference between a 3.3V and 5V Pro Mini is the regulator.
Are these assumptions correct? If yes, can I use a 5V Pro Mini like a 3.3V by powering it with a 3V Battery on the 5V pin?

user30488: First I am going to say it is very much possible. Allow me to explain.
The 5v version can be powered with 3.3v, but it is not recommended to supply power through the 5v pin, bypassing the regulator. Only regulated 5v power should be supplied through the 5v pin. Any other voltage levels should go in the RAW pin. If you supply the Arduino Pro Mini 5b with 3.3v the chip will only be underclocked to 8-12MHz, and the voltage for Vcc (the board's operating voltage and what it will power its pins with) will be 3.3v; but there is another way.
I primarily source my components from SparkFun due to their reliability, cheap price and awesome support. They provide the Arduino Pro Mini in 3.3v and 5v variants for only $10. I have used both variants for many purposes, but primarily when working with embedded and small solutions. One of those times I was accessing sensors that ran on 5v TTL logic, but I didn't have room to place a >5v battery, only room for a LiPo (which generally powers 3.3v).
I was able to solve this issue with a SparkFun 5v Step-Up Breakout*, which boosts the voltage to 5v without any other major issues. I should warn you that there are some issues you could face if you are using a LiPo to feed into the Step-Up. Even if you aren't I'll list it here for reference. The Step-Up's maximum safe input voltage is 4v, and can be a bit unreliable if the voltage goes above it. Unfortunately a LiPo's discharge curve places it above 4v when it is within the upper 95% soc (state-of-charge), so a 3.3v regulator (which SparkFun also sells) would be needed to reliably power an Arduino Pro Mini 5v through a Step-Up breakout (at least in my experience). Also, power from a Step-Up breakout is regulated.
*Pololu also sells a comparable Step-Up that could handle an input up to 5.5v in theory, but I haven't personally used it.

user7924: Im feeding 5V pro mini clone with 2 AA battery, to VCC pin, about 3V and it works normally. It consumes few current compared to 5V.  
And you can see what it the setup in here.
Also this video can be useful.

user4359: Pro Mini 5V has voltage regulator for 5V and 16MHz quartz.
Pro Mini 3.3V has voltage regulator for 3.3V and 12MHz quartz.
According to manufacturer specification Atmega328 chip powered by 3.3V cannot run 16MHz and needs at most about 13MHz. But it's official specification. Some chips are better than specification (faster) some are worse (slower). So you can try whether your Pro Mini is better or not. Or you can change quartz to 12MHz, if you can solder SMD parts.
Or better look for cheaper 3.3V clone, they are not much more expensive as 5V version...

user13573: I just discovered that I knew the answer to this, because I was accidentally running some 5v/16MHz Pro Mini boards at 3.3v without issue.  
Short Version
Yes, you can run one feeding it 3.3v, but you need to program it as a 5v/16MHz board to get the clock speeds right.  Also, all your TTL levels (outputs) will be 3.3v, not 5v.  Additionally, at least in my experience, the processor will run at 16MHz and not 12MHz with 3.3v.
Background Story
I had some generic Pro Mini boards that didn't mark their voltage at all, so I tried programming (and running) them using my FTDI at 3.3v using the 3.3v/8mHz option in the Arduino IDE.  I was able to upload programs just fine, but the clock was clearly wrong, as the "Blink" program ran about 2x as fast.  That told me that it was running faster than 8MHz .
So I tried again this time telling the IDE it was a 5v/16MHz chip, and everything ran fine, despite the fact that I was feeding it 3.3v.  I even saw that pin 13 was cycling from 0v up to 3.3v - exactly what I was feeding it.  So for a while I just thought they were 3.3v Pro Minis.  
But then thinking about it, I realized that having to tell it to run at 16MHz to get the clock right meant it really was a 16MHz chip - which I didn't think was available with the 3.3v model.
So on a whim (and since I have a ton of these little boards), I changed my FTDI to 5V, and was able to run and program the board just fine.
So from that, I conclude that yes, you can run them at 3.3v, but your TTL levels will be just 3.3v.

4661: <p>I want to run the 5V board at 3.3V because I can get the 5V version cheaper than the 3.3V version.</p>

<p>I read that you can power the Arduino Pro Mini with a battery by connecting the battery to the 5V pin, bypassing the regulator.</p>

<p>Somewhere else I read that the only difference between a 3.3V and 5V Pro Mini is the regulator.</p>

<p>Are these assumptions correct? If yes, can I use a 5V Pro Mini like a 3.3V by powering it with a 3V Battery on the 5V pin?</p>
 30488: <p>First I am going to say <strong>it is very much possible.</strong> Allow me to explain.</p>

<p>The 5v version can be powered with 3.3v, but it is <strong>not recommended</strong> to supply power through the 5v pin, bypassing the regulator. <strong>Only <em>regulated</em> 5v power should be supplied through the 5v pin. Any other voltage levels should go in the RAW pin.</strong> If you supply the Arduino Pro Mini 5b with 3.3v the chip will only be underclocked to 8-12MHz, and the voltage for Vcc (the board's operating voltage and what it will power its pins with) will be 3.3v; but there is another way.</p>

<p>I primarily source my components from <a href="https://www.sparkfun.com" rel="nofollow noreferrer">SparkFun</a> due to their reliability, cheap price and awesome support. They provide the Arduino Pro Mini in <a href="https://www.sparkfun.com/products/11114" rel="nofollow noreferrer">3.3v</a> and <a href="https://www.sparkfun.com/products/11113" rel="nofollow noreferrer">5v</a> variants for only $10. I have used both variants for many purposes, but primarily when working with embedded and small solutions. One of those times I was accessing sensors that ran on 5v TTL logic, but I didn't have room to place a >5v battery, only room for a LiPo (which generally powers 3.3v).</p>

<p>I was able to solve this issue with a <a href="https://www.sparkfun.com/products/10968" rel="nofollow noreferrer">SparkFun 5v Step-Up Breakout</a>*, which boosts the voltage to 5v without any other major issues. I should warn you that there are some issues you could face if you are using a LiPo to feed into the Step-Up. Even if you aren't I'll list it here for reference. The Step-Up's maximum safe input voltage is 4v, and can be a bit unreliable if the voltage goes above it. Unfortunately a LiPo's discharge curve places it above 4v when it is within the upper 95% soc (state-of-charge), so a 3.3v regulator (<a href="https://www.sparkfun.com/products/526" rel="nofollow noreferrer">which SparkFun also sells</a>) would be needed to reliably power an Arduino Pro Mini 5v through a Step-Up breakout (at least in my experience). Also, power from a Step-Up breakout is regulated.</p>

<p>*Pololu also sells a <a href="https://www.pololu.com/product/2564" rel="nofollow noreferrer">comparable Step-Up</a> that could handle an input up to 5.5v in theory, but I haven't personally used it.</p>
 7924: <p>Im feeding 5V pro mini clone with 2 AA battery, to VCC pin, about 3V and it works normally. It consumes few current compared to 5V. <a href="https://i.stack.imgur.com/CwZiJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CwZiJ.jpg" alt="consuming at 3V"></a> <a href="https://i.stack.imgur.com/Lb51E.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Lb51E.jpg" alt="at 5V"></a>
And you can see what it the setup in <a href="https://twitter.com/anunidentified/status/1069283092135845888?s=19" rel="nofollow noreferrer">here</a>.
Also <a href="https://youtu.be/381Rg5YDmRM" rel="nofollow noreferrer">this</a> video can be useful.</p>
 4359: <p>Pro Mini 5V has voltage regulator for 5V and 16MHz quartz.
Pro Mini 3.3V has voltage regulator for 3.3V and 12MHz quartz.</p>

<p>According to manufacturer specification Atmega328 chip powered by 3.3V cannot run 16MHz and needs at most about 13MHz. But it's official specification. Some chips are better than specification (faster) some are worse (slower). So you can try whether <strong>your</strong> Pro Mini is better or not. Or you can change quartz to 12MHz, if you can solder SMD parts.</p>

<p>Or better look for cheaper 3.3V clone, they are not much more expensive as 5V version...</p>
 13573: <p>I just discovered that I knew the answer to this, because I was accidentally running some 5v/16MHz Pro Mini boards at 3.3v without issue.  </p>

<p><strong>Short Version</strong></p>

<p>Yes, you can run one feeding it 3.3v, but you need to program it as a 5v/16MHz board to get the clock speeds right.  Also, all your TTL levels (outputs) will be 3.3v, not 5v.  Additionally, at least in my experience, the processor will run at 16MHz and not 12MHz with 3.3v.</p>

<p><strong>Background Story</strong></p>

<p>I had some generic Pro Mini boards that didn't mark their voltage at all, so I tried programming (and running) them using my FTDI at 3.3v using the 3.3v/8mHz option in the Arduino IDE.  I was able to upload programs just fine, but the clock was clearly wrong, as the "Blink" program ran about 2x as fast.  That told me that it was running faster than 8MHz .</p>

<p>So I tried again this time telling the IDE it was a 5v/16MHz chip, and everything ran fine, despite the fact that I was feeding it 3.3v.  I even saw that pin 13 was cycling from 0v up to 3.3v - exactly what I was feeding it.  So for a while I just thought they were 3.3v Pro Minis.  </p>

<p>But then thinking about it, I realized that having to tell it to run at 16MHz to get the clock right meant it really was a 16MHz chip - which I didn't think was available with the 3.3v model.</p>

<p>So on a whim (and since I have a ton of these little boards), I changed my FTDI to 5V, and was able to run and program the board just fine.</p>

<p>So from that, I conclude that yes, you can run them at 3.3v, but your TTL levels will be just 3.3v.</p>
