This has come up before, but nobody has offered any clarification.
If you look at the Tronixlabs tutorial picture of the two modules, you'll see that the microSD module at the bottom has a voltage translator chip that converts the Clk, MOSI and CS lines from 5V to the 3.3V needed by the SD card. But the full-size SD module only has inline resistors which leave the voltage at 5V on those lines, but limit the current.
You see these inline resistor versions used a lot with 5V Arduinos, and I don't understand how they could possibly work dependably. Some cards may deal successfully with the over-voltage state, but it's hard to see how they all could. I mean, you are applying voltage to those three lines that exceeds the absolute maximum ratings of the SD card when it is powered by 3.3V, as it should be. (Both modules do have the 3.3V regulator, so I assume power to the SD card is correct.)
So if you read through here, you'll see lots of people with the same problem you have - for some unknown reason, it just doesn't work. I just wish someone knowledgeable about the use of SD cards with Arduinos would explain how these goofy SD modules can work without voltage translation. It would also be useful to know if things do work correctly if you switch from an inline-resistor module to a voltage-translator module, or whether it still just doesn't work.
Of course the other possibility is that you just have the lines connected wrong - because things are different on the Leonardo, or whatever. I just don't know about that. Sorry not to offer any solution, but maybe someone else will see this and bail us out.