What is the difference between 賺錢 and 掙錢？ Today I went to a language exchange and a nice lady from Beijing told me to use 掙錢 instead of 賺錢。 I asked whether the latter was 口語 and she replied 賺錢 was something one said “just like that”. So now I am confused. 

For 掙錢 pleco defines: earn (or make) money
For 賺錢 make money (or a profit)

The first one does sound a bit more like involving work, but I cannot really be sure of the distinction.
I'd like to make a supplement based upon previous answers:

Usually, 挣钱 and 赚钱 are exactly the same. But in some special cases, 挣钱 could reflect a scenario of hard working, since making money is not easy, while 赚钱 could mean that making money is easy, compared with 挣钱. 

Here, the verb 挣 sounds like you have to make great effort to, you can see the verb 挣扎, which means struggle. The verb 赚 means making a profit, it can also be used as an adjective.

So we have the following phrases: 努力挣钱，working hard to make money (though 努力赚钱 is also possible) and 躺着赚钱, making money while lying on the bed.