Barrow's grease was a fairly standard term for lard until the mid-1600s, and continued to be used into the early 1700s. This is somewhat odd, as lard was also a common enough term from the mid-1400s on. There may have been a perceived difference in quality or an actual difference in cost, but I have not been able to find evidence of either.
Grease was a general term for oils and fats; white grease, for example, meant butter. A barrow was a pig that had been castrated, to avoid boar taint. While this might suggest that barrow's grease was particularly that grease to be used for cooking, lard was also used for cooking, and both were also used in medicines and makeup.
To take away the scars of the small pox
Take barrow's grease a pretty quantity, and take an apple and pare it and take the core clean out, then chop your apple and your barrow's grease together, and set it over the fire so that it may melt but not boil, then take it from the fire, and put it thereto a pretty quantity of rose water and stir all together till it be cold, and keep it in a clean vessel, and then anoint the face therewith.
The English Huswife by Gervase Markham, 1615.
The continued use of the term barrow's grease was surely largely due to local dialects, but if there was any additional difference between it and lard, I have not yet been able to find it. By the 1700s it was largely used only in certain dialects, and has today fallen entirely into disuse.
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