Question about the difference between a tsunami caused by a meteorite and one caused by an under water earthquake Picture the situation where a meteorite falls somewhere on one of our oceans and gives off all its kinetic energy to the water (let's ignore the thermal energies involved like, for example, the heat caused by the atmospheric entrance friction). 
Now imagine that below the same place as where the meteorite hits the water (on the bottom of the ocean) a big earthquake occurs which gives off the same amount of energy to the water as the meteor did. 
Let's further assume that the waves caused by both events will have noticeable effects when reaching landmass. 
The earthquake lifts the water up from beneath, while the meteorite pushes the water down from above, like a droplet falling into a pool of water.
Will the effects on the same piece of landmass land have the same devastating force, the only difference being that in an earthquake tsunami the ocean will first withdraw, after which the big wave enters, while in the meteor tsunami the big wave will directly hit the landmass? Or are there differences which make one of the two more devastating than the other?
 A: The tsunamis created by a meteor impact versus tectonic activity are very different.
Meteor and/or Landslide Tsunamis
Tsunamis generated by meteor impacts or landslides produce an, initially, very tall (relative to surrounding water surface) wave that propagates in a roughly circular manner away from the source.  Due to the high initial wave height and generally smaller amount of total displacement (compared to earthquakes), the wave that hits shore (assuming meteor impacts in center of ocean) is much shorter and generally has less total volume.
However, this does not necessarily make the devistation any less significant.  For instance, a 30 meter wave crashing down on top of you will immediately crush you (it would probably crush a 1000 foot oil tanker too).  Because of the tremendous initial heights of these types of tsunamis, they have been called megatsunamis.
Note that the maximum height of the resulting wave from a meteor impact in open water cannot exceed the local water depth [e.g., see Melosh, 2003].  Further, large impact-like tsunamis generally have a tall initial peak that quickly suffers from wave breaking that greatly reduces the height [e.g., Hofmann et al., 2007; Wunnemann et al., 2010].
Tectonic Tsunamis
Tsunamis generated by earthquakes are generally small out at sea (i.e., in deep water) and do not typically produce large wave heights even when approaching shore.  In some cases however, like the 2011 Tohoku earthquake (over 100 feet in some places), the tsunami resulting from tectonic activity can have a large wave height at landfall.  The first thing observed is the trough, which appears as if the ocean is running away from the shore.  In most cases, what follows seems like a typical bore wave that has a semi-endless supply of water behind it.
Because the event stems from massive areas of land displacement, the total volume of water can be huge.

Now imagine that below the same place as where the meteorite hits the water (on the bottom of the ocean) a big earthquake occurs which gives off the same amount of energy to the water as the meteor did.

Meteor impacts large enough to create tsunamis with noticeable devistation can generally affect the Earth's crust as well [e.g., see Melosh, 2003].  Even if we assume the impactor is not large enough to excavate enough water to directly impact the ocean floor, the type of wave produced by the impact versus the earthquake is very different.

The earthquake lifts the water up from beneath, while the meteorite pushes the water down from above, like a droplet falling into a pool of water.

No, not really.  The impactor will displace water, which generally means it must move transverse with respect to the impacting objects velocity, not downwards.  Water is, for the most part, an incompressible fluid.  Thus, you cannot "push water down" without also pushing down the Earth below.

Will the effects on the same piece of landmass land have the same devastating force, the only difference being that in an earthquake tsunami the ocean will first withdraw, after which the big wave enters, while in the meteor tsunami the big wave will directly hit the landmass?

It is generally assumed that both wave types will follow the shallow water wave dispersion relation approximation so long as the impacting body does not excavate water all the way down to the ocean floor.  Thus, they should have similar propagation speeds.  However, the impacting body's wave will be a tall, breaking wave that loses a lot of energy very quickly [e.g., Hofmann et al., 2007; Wunnemann et al., 2010] whereas the earthquake-driven wave conserves more energy longer because it does not break.
The long answer is actually very complicated and part of ongoing research, which requires extremely detailed numerical simulations.
The short answer is that if a large meteor (e.g., >100 m) hits simultaneously and colocated with a large earthquake (e.g., M > 8), everyone is in trouble.

Or are there differences which make one of the two more devastating than the other?

If the impact occurs close to shore and the fault causing the earthquake is not a megathrust type, then the meteor-driven tsunami would cause more immediate damage along the shoreline due to its much larger wave height and crashing waves.  The long-term damage is difficult to estimate (e.g., water damage) without further specifics.
References


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*Hofmann, K., et al. "Oceanic Impacts - Types and Characteristics of Induced Water Waves," 38th Lunar and Planetary Science Conference, No. 1338, pp. 1586, 2007.  2007LPI....38.1586H

*Melosh, H.J. "Impact-generated Tsunamis: An Over-rated Hazard," 34th Annual Lunar and Planetary Science Conference, No. 2013, 2003. 2003LPI....34.2013M

*Wunnemann, K., et al. "The Impact-induced Tsunami Hazard — Insight from Numerical Modeling of the Eltanin Event," 41st Lunar and Planetary Science Conference, No. 1533, pp. 2220, 2010. 2010LPI....41.2220W
