Those are political slogans, not economics, despite “economics” in the name. Trickle down specifically was a pejorative used to criticize reducing tax burdens on corporations and wealthy investors. Supply side isn’t a clear set of policies, but presumably they are policies that target aggregate supply (production capability) of the economy but again is largely a political term for policy to directly benefit businesses (suppliers). More on reddit.com

Why are there supply siders and demand siders but no "it depends" siders? In practice there are only "it depends" siders. Even the so-called "Supply side Economists" (who were a fairly small group) were only saying that the demand side had been overemphasised by Economists before them. Everything in economics is about the interaction of supply and demand. All theories that aim to deal with any significant part of the economy deal with both. Different limitations apply to different parts of the economy at different times. It's not really the same as Liebig's law or the rate-determining-steps in Chemistry. There are some similarities though. More on reddit.com

This comes up a lot in economic debate and the problem is that reducing any complex reality to a simple rule will, at best, produce a useful guideline and, at worst, lead to disastrous results. "Trickle down" is more accurately called "supply side economics" because the idea is that more production equals more wealth. That's actually a oretty useful guideline, but it's not an absolute rule. It assumes that there is always latent demand and that if we can produce more, we can create more jobs and more consumption. Stimulating growth becomes about putting more resources into the hands of businesses and entrepreneurs and getting out of their way. Naturally, this approach favours savings and investment to increase productive capacity rather than consumption. The opposite, "trickle-up" economics, is more accurately called "demand side economics" amd the idea is that more consumption equals more wealth. The assumption is that there is always latent productive capacity and that if we can simply stimulate demand, we can increase consumption, jobs, wealth, and use more of our productive capacity. This is an excellent guideline during crises, recessions, and depressions, but not as useful during full-employment when there is little or no unused productive capacity. Naturally, this approach favours consumption at the cost of reducing savings and investments. The idea is to put cash or credit in the hands of those who will spend, not save. Economics, however, is about supply and demand. Both "trickle-down" and "trickle-up" are extremely useful guidelines in different situations, but only looking at one or the other is like trying to drive your car using only the gas pedal or the break pedal. You absolutely need both and it's better to frame them as supply and demand rather than trickle up or trickle down. More on reddit.com

I believe trickle-down is a more specific set of policies within the larger realm of "supply-side economics". While supply-side fiscal policies focus generally on increasing aggregate supply, the trickle-down approach argues that this is better achieved by focusing on policies that favour the upper class and large corporations. For example, a "supply-sider" can advocate lowering taxes in a progressive manner, taxing everybody less but the poor even lesser. While the trickle-down approach would preffer lowering taxes in a regressive manner. More on reddit.com