I included TempLS with GHCN adjusted data as well. This gives the opportunity to also show the differences between adjusted and unadjusted, and to plot the actual differences.
Nothing looked unreasonable. The big Jul-Aug difference between GISS and TempLS was in the treatment of Arctic (especially) and Antarctic. There is actually very little effect of GHCN adjustment. The month-month changes are themselves quite interesting; the patterns are quite persistent.
The GISS data I used was the 1200 km interpolated, (download gistemp1200_ERSSTv4). The TempLS runs used ERSST4 and GHCN from here. I fitted each set with spherical harmonics up to L=10 (121 functions). This gave enough resolution to make the comparisons, without risking artefacts.
In this tableau, for each set the July 2015 result is on the left, and August on the right. All data uses base years 1951-80, with conversion numbers from here.
Giss shows the Arctic going from part-warm to moderate everywhere. TempLS, however, has the Arctic warm in each month. In both datasets, East Antarctica is cold in both months, but generally a larger area in August. TempLS shows greater cooling in the East US. Both show the August warmth in S Americe, which is in fact the most noticeable change over the month.
You see a slight variation in SST with adjustment, which may seem odd, as only land is adjusted. But the spherical harmonic approx is global, so the effects can spread. The difference is small.
The next plot shows the actual differences Aug-July.
This emphasises the polar changes in GISS, with much less Antarctic change in TempLS, and almost none in the Arctic. A diffreence is not surprising, as GISS uses extra data in the south, and their interpolation scheme in the Arctic is also different. The other main differenve is in the US, where TempLS emphasises both the Eastern cooling and Western warming.
The next tableau shows the datasets differences, just for August.
Very little effect of GHCN adjustment, except for some in Antarctic. This is expected, because present time is the reference point for adjustment. The differences will be due to changes in the offsets. Between GISS and TempLS, you see another view of the polar differences, and otherwise just odd patches, mostly over land.