The similarities are of course the economic recession through which the country has just suffered. Also, the causes are similar; greed not only by investors, but others, who pushed the economy to the brink to make a fast dollar. The primary difference is that there are now safeguards in place which did not exist then, so the effect of the downturn is not as traumatic.
During the Great Depression, there were no safeguards to prevent wild speculation in the Stock Market. One could purchase stock on margin (investment loan) for as little as 5% of the purchase price. Now the required margin is 50%. During that time, the Dust Bowl resulted because farmers did not rotate crops or use good soil preservation techniques. Farmers have since learned not to make that mistake. Also there was no insurance in the event of bank failure. When banks failed, depositors lost everything. The very rumor that a bank might fail resulted in a "run" on the bank (all depositors demanding their money), which only served to make it fail that much more quickly. The Glass Stegall Banking Act created the Federal Deposit Insurance Corporation, since which time, no depositor has lost money in a federally insured bank.
Finally, Social Security is now in existence so the elderly, disabled, and widowed/orphaned have some means of subsistence. This was also not true during the time of the Depression.
The present Recession has been painful and expensive; however the American people have not lost hope as they did at that point. One other similarity you might consider. The Presidents who dealt with both were Democrats, and both believers in Keynesian economics--massive government expenditure to fight the downturn.
For the most part, there is no comparison between the Great Depression and today. In just about every way you can think of, people today are so much better off than those who lived during the Depression.
The major difference can be seen in the unemployment rate. Today, we are upset because the rate is hovering close to 10%. During the Depression, it was anywhere from 25% to 33% nationwide. We're not even close to how unemployment was back then.
We also do not see anything like the kind of poverty that existed back then. There aren't lines at soup kitchens. There aren't people out selling apples on the streets. There aren't a ton of people trying to move to California in hopes of getting work like the "Okies" did.
While things are way better now than during the Depression, there is one thing that could be similar. That is the fact that the rich are getting richer and the poor are getting poorer. In the Depression, the rich had a huge share of the nation's wealth--there was a great deal of inequality. That inequality went down for a while, but now it is back--the richest few people hold a much larger percentage of the wealth in the US than they have until recently. In that way, the economy is sort of like how things were during the Depression.