Protected 3.7v battery as replacement for 3.8
NB: This is a "best attempt" answer.
It is very probably correct but is offered on a no responsibility YMMV basis.
In your position I'd take my advice :-).

IF these are Lithium Ion cells then the 3.8V and 3.7V ratings are nominal mean voltages. A LiIon cell is at about 4.2V fully charged dropping to about 3V when fully discharged. 

Charge the existing battery (if still working).
If Vbat unloaded is about 4.2V you (almost certainly) have a LiIon cell and the 3.7V cell will (almost certainly) work perfectly well.   

Having the same mAh rating is desirable.
A higher mAh new cell will take longer to charge BUT will OVERcharge somewhat due to the way LiIon charging works. 

Photos of old and new cells and specs such as are known will help .