The basic differences:

ASTM A615 is the standard for reinforcement bars in concrete building structures.  The most common grade is Grade 60 (min. yield strength of 60 ksi, min. tensile strength of 90 ksi) but it is also available in Grade 40 and Grade 70.  A615 has no upper carbon limit, so the C.E. (carbon equivalent) of A615 is not controlled and must be determined through chemical analysis.  This analysis is shown on the MTR.  Some MTR's will actually show the C.E. in addition to the chemical analysis, but it must be calculated if it's not shown.

ASTM A706 is the standard for reinforcement bars for applications requiring controlled tensile properties and applications requiring enhanced weldability.  It is only available in Grade 60 (min. yield strength of 60 ksi) and the min. tensile strength is 80 ksi.  The C.E. of A706 is controlled to provide a carbon equivalent not to exceed 0.55%, but like A615, the C.E. may or may not be shown on the MTR.  If not shown, it will need to be calculated.  

All A615 bars are marked with an "S" and all A706 bars are marked with a "W".  Bars that are marked with both an "S" and a "W" are dual cert. bars.

The weldability of reinforcing bars is primarily based on the carbon equivalent content (C.E.).  The carbon equivalent content of steel refers to method of measuring the maximum hardness and the weldability of the steel based on the chemical composition of the steel.  Higher concentrations of carbon (C) and other alloying elements such as manganese (Mn), chromium (Cr), silicon (Si), molybdenum (Mo), vanadium (V), copper (Cu), and nickel (Ni) tend to increase the hardness and decrease the weldability of the material.  Each of these materials tends to influence the hardness and weldability of the steel to different magnitudes, thus making a method of comparison necessary to determine the difference in hardness between steels made of different alloying elements.  The C.E. content is the most common method of comparison, but others exist, such as the nickel equivalent content and the chromium equivalent content.

The calculated carbon equivalent is compared to the minimum preheat and interpass temperature Table 5.2 in D1.4  This temperature is based on the C.E., the reinforcing bar diameter, and the surrounding temperature.  Preheat and interpass temperatures of Table 3.2 in D1.1 must also be followed when the bars are to be welded to common structural steels such as A36, A500 B, A53 B, A572 Gr. 50, A992, and others.  The higher minimum preheat and interpass temperature of the two tables is the one that's to be used during welding.

Based on certain C.E. percentage ranges, here's what can be expected:

C.E. % < 0.14    Excellent weldability.  No special precautions are necessary

C.E. % 0.14 <0.45    Martensite, the hardest microstructure that can form in steel, is more likely to form.  Modest preheats with low hydrogen electrodes become necessary.

C.E. % > 0.45    Extreme complications.  Weld cracking is very likely unless proper preheat is applied.  Low hydrogen electrodes are required.

More info in D1.4