They're both a solution to the same problem: the lack of money movement in the economy. They both try to address it by pumping money, disposable income, and hoping the money starts moving again.
The difference is that government can direct specifically where those funds are to be used (ie: infrastructure projects, science projects, etc), and thus have a direct impact on jobs, disposable income and the overall economy. Plus government doesn't have an intrinsic goal of making money. Sure, in the micro they pick winners and losers, and the winners certainly make money, but the overall goal is in the macro. The criticism with it is that government allocation of funds is inefficient and crony, and that government is slow to retract from the market once the economy is moving again.
Trickle down was essentially the same thing, addressing the inefficiencies and the picking of winners and losers. It just stopped working because companies stopped caring about the country's economy, and prioritized their bottom line instead. It intrinsically trusts the beneficiaries of actually reinvesting the extra money in Main Street, but Wall Street is more profitable, incurs less risk, and there's really nothing preventing companies from hoarding the cash. So unless you actually make Wall Street a less desirable investment than Main street, this is not a workable solution anymore.
They're both interventionist solutions to deal with a very specific problem. You can get away with monetarism and moving the interest rate levers while the economy is working fine, but once that doesn't cut it, you're going to need government intervention in one way or the other.
This whole thing is certainly more complicated than two dogmas, and encompasses the investment markets (especially derivatives), the economic realities (deflation, inflation), etc.