During the time period that you mention, the economy of the North became much more market-oriented, much more industrialized, and much bigger. These changes did not happen to nearly such a great extent in the South.
First of all, the Northern economy became more industrialized. In 1800, there were essentially no machines being used in the Northern economy. As the years went by, however, this changed. In particular, industrialization created a textile industry in the Northeast and it created a network of railroads in the North by 1850. There were also other factories of various sorts that sprang up in the North over this time period.
Second, the Northern economy expanded. Large numbers of immigrants came to the North, particularly in the 1840s. The Northern economy also expanded with the creation of better transportation. This started with the building of the Erie Canal (completed in 1825) and then continued with the invention of steamboats and, most importantly, the creation of a railroad network. This made the North much richer than the South by 1850.
Finally, the Northern economy was much more market-oriented by 1850. In 1800, most Northerners had produced most of what they needed and had bartered for the rest. People essentially depended only upon themselves and their neighbors and had very little use for cash. As time went by, however, the Market Revolution occurred. This was a process where more people started working for wages (as in the new factories) or started growing crops to sell rather than to use for themselves. Over this time period, then, the Northern economy changed from being based on subsistence farming and minor bartering to being based on selling goods or labor for money and using that money to buy things that you needed.
In these three major ways, the Northern economy changed from 1800 to 1850.