Yes it is normal, though I am unsure as to what the reason for the  difference between teh two methods is. Nanodrop values are often higher compared to PicoGreen. The latter is more reliable for the later steps of the experiment.

So even a 10-fold difference is normal?

Nanodrop measures impurities and degraded DNA as well at 260nm. Picogreen only binds to dsDNA so that's the difference. For sensitive applications and low concentration of DNA it is better to use a fluorescent based DNA concentration assay. In my experience, the Nanodrop overestimates genomic dsDNA up to 20-25% as compared to the Cubit dsDNA BR assay. For

So even a 10-fold difference is normal?

10-fold difference seems a lot but it is possible. One thing that can affect the readout from Nanodrop is if airbubbles are trapped in the measurement column. Make sure to load at least 1.5ul sample volume.

I recently compared Qubit and Nanodrop for quantifying genomic extracts from insects. Qubit works differently from picogreen, but offers better accuracy and sensitivity compared to Nanodrop. The Nanodrop readings (~50 ng/ul) were nearly 10x higher than the Qubit readings (~5 ng/ul). This was a small-scale test, with only five samples. I was a little stunned by this result. Either the DNA was highly contaminated or the the applications of the tools were erroneous. I plan to perform additional tests on more samples.  

One person reports here that "the nanodrop usually gives a number that is 5X-10X higher than the Qubit reading." http://seqanswers.com/forums/showthread.php?t=31560&highlight=Qubit+Nanodrop 

Another similar post: "I am with Harold here. Nanodrop is not only useless at this concentration, it is likely to give you and incorrect result even at 10x this concentration." http://seqanswers.com/forums/showthread.php?t=24386&highlight=Qubit+Nanodrop

I have compared NanoDrop and PicoGreen in 175 samples. I have found that NanoDrop readings were about 4 times higher than PicoGreen. I now trust PicoGreen!