Error when calculating Alt/Az from Ra/Dec
The algorithm on that page is very simplistic and is probably the source of a good portion of that error.  It's best to test your code using a very accurate ephemeris, even if your final goal is to have a much more simple ephemeris implementation.  DE405, or VSOP87 would be good alternatives.

A small error in GMST can have a big effect on Alt/Az computations, I would expect that the simplicity of the algorithm on the page produces a good sized error in GMST.

Another common source of error is differences in time scales.  You need to make sure you propperly account for things like leap seconds, which that page specifically says it ignores.  You need to convert UTC to Terrestrial Time by subtracting 37.0 + 32.184 (where 37 is the current number of leap seconds).

Also make sure you're not comparing different coordinate systems.  Like J2000 vs "Of Date", or coordinates adjusted for atmospheric refraction.