If heat is merely molecular motion, what is the difference between a hot, stationary baseball and a cool, rapidly moving one? This is from the  Exercises for the Feynman Lectures on Physics, exercise Exercise 1.1. I believe that a hot stationary ball has more thermal energy due to the inter-molecular motion of the baseball, while a cool, fast moving baseball has more kinetic energy due to the macro-object itself moving faster in a particular direction?
 A: The difference is the manner of the motion. The motion of the ball when thrown is highly (maximally, in fact!) organized, while the thermal motion contributing to its temperature is highly 
(also maximally, in an analogous sense) disorganized. Or, to say it another way, kinetic motion looks the most like signal, while thermal motion looks the most like noise: think about the static on an old TV monitor when tuned to a dead channel, versus when tuned to one with a movie - that is "essentially" the exact same difference as between the motion of your thrown ball and the same ball at temperature.
There are a number of different ways you can formalize this notion mathematically, that have slightly different properties and measure it in different ways - Shannon entropy, Kolomogorov complexity, etc. - but basically they all boil down to "how much something resembles what you are most likely to get from nothing but chance alone". 
Think about it like this: Suppose you were to take each molecule, one at a time, within the ball. You are going to assign them, one at a time, a motion. The total energy of those motions you are going to give them is fixed in advance. Suppose you choose the direction they should each be moving completely at random, as well as what amount of energy they get from the total. You should expect that all the motion vectors should be pointing every which way on each and every molecule. That's heat.
Suppose you now do the same, but you pick all the vectors to have the same size and point all in the same direction. That's kinetic motion (thrown ball). Now in theory, you could get that by chance, but it would require phenomenal luck: for all of, say, $10^{24}$ random pickings, they'd all have to point in more or less the same direction! It'd be like playing and winning a lottery ticket with $10^{24}$ "lotto balls" versus playing and winning one with only six!
A: You need to provide the context of Feynman as I'm having difficulty in understanding the statement as you've written it. I'll attempt an answer assuming I'm interpreting it correctly.
To begin with, the statement in the title, "heat is merely molecular motion" is incorrect. Heat is energy transfer between objects due solely to temperature difference. Heat can be thought of as a mechanism by which the internal kinetic energy of a cooler body is increased by transferring some internal kinetic energy from a hotter body. But heat is not the molecular motion (kinetic energy) itself. That is properly called internal energy. The temperature of the baseball is a measure of the average translational kinetic energy of the molecules that make up the baseball as a result of their random motion. It's a measure of its internal microscopic kinetic energy.
The kinetic energy of the molecules of the thrown baseball to their collective, i.e., unidirectional motion, is its macroscopic kinetic energy. It has no bearing on the internal energy (temperature) of the baseball, and the internal energy of the baseball has no bearing on the collective macroscopic kinetic energy of the molecules of the baseball.
So with the above distinction in mind, a "hot" stationary baseball will have higher internal kinetic energy than a cold thrown baseball, because the two forms of kinetic energy are independent of one another.  The internal kinetic energy of the baseball depends only on the temperature. The kinetic energy of the baseball as a whole depends only on its speed.
You may find it interesting to compare the hypothetical increase in the internal molecular kinetic energy of a "hot" baseball as a result of heating, to the increase in macroscopic kinetic energy of a cold thrown baseball due to being thrown (due to Work).  Here's a hypothetical example.
Macroscopic Kinetic Energy of Thrown Baseball-
Let's first consider the macroscopic kinetic energy of the cold thrown baseball, "cold" meaning room temperature, say 25$^0$C. A regulation baseball has a mass of 0.145 kg. The fastest speed recorded of a pitch is about 105 mph, or 160 km/hr (Aroldis Chapman of the San Diego Padres). The macroscopic kinetic energy attained by the baseball is
$$KE=\frac{mv^2}{2}$$
Plugging in the numbers that's a kinetic energy of about
$KE$ = 0.16 kJ.
In other words, the amount of energy needed in the form of work (throwing the ball) to increase the speed of the ball to 169 km/hr is about 0.16 kJ.
Microscopic Kinetic Energy (Internal Energy) of a Heated Baseball-
Now, let's consider the energy (heat) required to raise the temperature of the ball (increase its internal kinetic energy). A baseball is made up of various materials. I could find no information on its specific heat.  For arguments sake, lets say it is rubber. The specific heat of rubber is about 2 kJ/kg.$^0$C. I'm not sure what is meant by "hot", but lets say we heat the baseball to 50$^0$C from 25$^0$C, for a 25$^0$C rise, which is not particularly "hot" for a non-metallic material. The required heat is
$$Q=mC\Delta T$$
Plugging in the numbers, we get
$Q$ = 7.25 kJ.
So, if I did my math correctly, it would take about 45 times more energy input to elevate the temperature of the baseball by 25$^0$C than it does to increase its velocity by 160 km/hr.
Hope this helps.
A: Definition of temperature in statistical mechanics terms:
$$T_{\text{kinetic}}=\frac{2}{3k}\left[\overline{\frac 1 2 m v^2}\right]=\frac{2}{3k}\text{KE}_{\text{average}}$$
Please read the linkfor the constants, what is important to note is that temperature is analogous to average kinetic energy
Also 3. applies 

When kinetic temperature applies, two objects with the same average translational kinetic energy will have the same temperature. 

So it is necessary  by the calculation  of temperature using kinetic energy statistically,  that the same inertial frame should be used . In thermodynamic terms, a thermometer measuring the temperature of the stationary hot ball would give the same reading if the ball is  in motion.
A: Thermal energy $is$ kinetic energy. Its the kinetic energy of random, molecular motion, including oscillatory vibrations, rotations, and any movement of the molecules within the material.
We can still talk about "thermal" and "kinetic" energy (notice quotes). For example, we can define "kinetic energy of the baseball" as the kinetic energy of the center of mass, and the "thermal energy of the baseball" as the kinetic energy of all the individual atoms, less the "kinetic energy of the baseball". In that case, then yes, your idea is correct.
However, it might be useful to think about the relative scale of these two components of kinetic energy. The energy of a single (average) atom in the baseball is something like $k_BT$, where $k_B$ is the Boltzmann constant. You could find the average velocity of a particular atom in the baseball, and you would find that it is much faster than you or I could throw it. A hotter ball has more "thermal energy", but it also has more energy overall. Thats why when you throw a baseball hard at the wall, it doesn't burst into flames.
A: The molecules of the hot stationary ball have more kinetic energy than the molecules of the cold moving ball because the stationary ball is hot -> when we touch it we feel our hands warm-> energy is overall transferred from the ball to us and the moving ball is cold -> when we touch it energy is overall transferred from us to the moving ball . 
A: This question is more profound than it might seem at first glance.  If the thermal energy of molecular motion is invariant and the kinetic energy of condensed matter motion isn't, it suggests that these two kinds of motion are not fundamentally similar.  Pushing the logic to its extreme, it requires that temperature remain invariant even at velocities approaching the speed of light.  But that would require that the molecular motion exceed the speed of light - which breaks physics.
It seems that there's something fundamentally wrong with the idea that molecular motion and kinetic motion are essentially equivalent but just happening on different scales.
