I can't install tiny core plus on usb

[WtMelplay]

I use rufus so that I can run tiny core, I'm very afraid of mixing up with the hdd or ssd flash drive, because there is important data there. I even used formatting, but the data on the flash drive did not change. I checked other disks, but they are also fine. And for some reason my hdd works as a flash drive (that is, it uses files from the flash drive). I'm a newbie, so I don't know what's best.

[Juanito]

I've never used rufus, but as an alternative, if you have access to a machine running linux you can manually install tinycore.

Otherwise, if you have access to a machine with a cd/dvd drive, you can burn the coreplus iso to cd/dvd, boot the cd/dvd and use tc-install to install tinycore to a usb stick.

[WtMelplay]

Unfortunately I don't have a CD drive, I just have a modern computer.

[CentralWare]

Rufus should be able to do the job without an issue.

1. Download the ISO file for your platform - based on your notes, I'm assuming x86_64
2. In Rufus, second line down, click the SELECT button and choose your downloaded ISO file
3. In the TOP line, be sure to select your USB drive - be sure it's the CORRECT DEVICE!
4. Click START at the bottom - once Rufus is done, be sure to properly EJECT the USB drive

Plug the USB drive/pen/stick/etc. into the machine you want to boot with Tiny Core and reboot it or turn it on.
1. Go into BIOS and be sure SAFE BOOT is in fact DISABLED; BIOS is usually the DEL key, F2 key or F10 key while it's booting up.
2. Be sure to select the USB flash drive as your main boot device
3. Save your changes and reboot - Tiny Core should now be booting

IN MY OPINION, if getting into BIOS and making changes causes any fear, you should probably have someone more familiar with computers doing all of this.

* If your device is an Andriod (such as Chromebook) or Apple (iMac/book/etc.) Tiny Core is not intended for those platforms at this point in time.
* If your device is Raspberry Pi, you should be downloading piCore instead of Tiny Core.

Finally,

I even used formatting, but the data on the flash drive did not change.

If you're uncertain as to what you're doing even within Rufus, you run the risk of wiping out everything.  Stop there and find someone more familiar with computers in general and have them assist you in person.  (Trying to talk someone through such things on the Internet never ends well.)  Even if you have to hire someone for an hour, it's better than accidentally erasing your hard drive - which you note, has information on it you don't want to lose.

Unfortunately I don't have a CD drive, I just have a modern computer.

Just in case someday you actually "needed" one, click here.  This is just an EXAMPLE of one of many, many outlets that carry USB based CD/DVD/BDR drives for cheap which is WHY computer manufacturers skimp out on as much as possible - to cut costs on THEIR end, pass some of those savings onto the consumer, and there's always a work-around should the consumer need such.
Remember Floppy Drives?  They have those in USB flavors as well.

Good luck!