There are some things you can consider, to get an idea of performance, absent specific gel tests and such. The numbers for the basic common ammo of each gives reference, as well.
The .44 Special factory load (standard, been made forever) is a 246gr lead at approx. 750fps.
.38 special is a 158gr @ 850fps.
The smaller .44spl guns are 5 shot, and you don't get 6 until you get to the larger, heavier "N" frame (S&W) size class. .38, you get 6 in the medium size guns.
.38 Special has a HUGE variety of different loads available, numerous bullet weights and designs, and the specialty loads (for suitable guns) of +p and even +P+ from some makers.
.44 Special is much more limited in its choice of factory ammo, but shines when handloaded.
.38 Special is one of the most widely available cartridges that exist, even if its not the top of the pyramid any more.
.44 Special isn't one of the top popular rounds, so it may not be common on the shelf stock everywhere.
Just a few things to consider..