The only dumb question is the one you don’t ask 
,
that said, no it isn’t identical to your original circuit, but that circuit likely won’t work. The raw pin has a 5v regulator connected to it, so the input voltage needs to be something like 7V or so (to a max of 12V I think.) A single cell Lipo battery is 3.7 to 4.2V (discharged to fully charged) so doesn’t have enough voltage to support the regulator on the raw pin. That is why you need the boost regulator on the charger board. Connecting the +Vout pin from the boost supply to raw on the pro mini, and adjusting the boost to be 7V will work, and then the HC05 will connect to VCC on the pro mini (which provides regulated 5V from the regulator on the pro mini.) As will setting the boost voltage to 5V and applying it to the VCC pin and the VCC of the HC05 (which is what I did here.) Your original circuit is also lacking any charging supply connection, you would need to supply a power source to the two right side pins of the boost module to charge the battery. Note that the new boost module has a micro USB port and if you connect a USB cable to that, then the USB port will provide the required charging power, as an alternative to the Vin pins (I wouldn’t use both, the boost module would hopefully protect the USB port, but may not in which case you may damage the USB port if the Vin voltage is higher than 5V.)
My original .fzz has only one, there is however an oddness (which seems to be a Fritzing bug perhaps) in that the pro mini doesn’t show in breadboard
The pro mini is actually there and will show up if you right click on the breadboard then click Raise and Lower->Send to back. This is a bug because breadboards are supposed to always be the lowest layer and not appear above anything else or at least so I was told when I started in Fritzing 5 years ago.) After clicking Move to back the pro mini re appears in breadboard
The pro micro is always there in schematic even when it didn’t appear in breadboard
edit use the correct image for schematic …
I would guess that you thought I had forgotten the pro mini and dragged one in which will indeed create two (on top of each other.) Deleting one of them should fix this up.
Peter