Can you give some examples which would reveal whether they are same things or not?
They're not the same thing because thermal equilibrium requires thermal contact. Two bodies can be at the same temperature, and not be in thermal contact, and therefore not in equilibrium. As I'll describe later, two bodies can also be in thermal contact, in equilibrium, but not at the same temperature.
Fundamentally, thermal equilibrium is all about the exchange of heat energy. When two bodies are exchanging heat energy, they are in thermal contact. When there is no net flow of energy between two bodies in thermal contact, they are in thermal equilibrium. Note the word "net". Even in equilibrium, when two bodies are in thermal contact energy will flow both ways between the bodies.
Funny thing, two bodies can be in thermal equilibrium and not be the same temperature. Take the atmosphere, for example. Now, the real atmosphere is really complicated, but we can consider a simplified example with no convection. In this example, the atmosphere at every altitude is in thermal contact with the atmosphere above and below it, right? Well, whenever a particle moves upward it loses some of its kinetic energy to gravity, and it gains when it travels downward. Because of this, the temperature in an atmosphere that is in local thermodynamic equilibrium will fall with altitude; all because gravity is able to tax the energy of up-going particles and rebate energy to the down-going ones.
Now, if by constant you meant in time, and not in space, then yes, thermodynamic equilibrium requires the temperature be constant with time.
Thermal equilibrium between two bodies implies that both bodies are at the same temperature. They can be in thermal equilibrium (i.e. having the same temperature) while the temperature changes due to some external energy input.
Constant temperature (isothermal), on the other hand, implies that temperature is not changing throughout some process.
Thermal equilibrium and constant temperature are therefore not the same thing.
First, you must start with a reference to an object with a temperature. That object is called the system.
When we say that a system has a constant temperature, we imply that the system also has a uniform temperature. A system with a uniform temperature has the same temperature throughout the entire system. When we want to describe a system with a temperature that is not uniform, we say so explicitly. For example, a system can have a constant linear temperature gradient. This is explicitly saying two things--the temperature is not uniform but it has a constant (linear) profile.
Constant temperature is otherwise taken to mean that the temperature is not changing with time.
In summary, when we say that a system has a constant temperature, we mean implicitly that the temperature is uniform throughout and we mean by direct inference that the temperature is constant in time.
In thermodynamics, a system is said to be in thermal equilibrium in one of two ways. It is in thermal equilibrium with itself when its temperature is uniform throughout. It is in thermal equilibrium with its surroundings when the temperature is the same in both the system and the surroundings. In both cases, the term thermal equilibrium is made in reference to something, either the entire system itself or the system and its surroundings.
In summary, when we say that a system is in thermal equilibrium, we mean by direct inference that its temperature is uniform throughout and we often also mean implicitly that its temperature is the same as the surroundings.
By this point we can recognize that when we say that a system is at constant temperature, we directly state that the system is in thermal equilibrium with itself and that it remains so over time. All that is left is to state explicitly whether the surroundings are the same temperature or not.
By this point we can recognize that when we say that a system is in thermal equilibrium (and say nothing else), we directly state that the system has uniform temperature throughout. All that is left is to state explicitly whether the system is in thermal equilibrium with the surroundings and to state whether this condition holds over time.
We may imagine a system where the temperature is not uniform throughout or a case where the temperatures of the system and the surroundings are different. In such cases, when no insulated boundaries exist, heat will flow. In thermodynamics, the first case (temperature differences inside the system) is not a case for true thermal equilibrium of the system. The second case (temperature differences between system and surroundings) is not a case for true thermal equilibrium between system and surroundings.
Those cases where we find that temperature differences yet also find that the temperatures remain constant in time are better said to be at steady state rather than to be at thermal equilibrium. Steady state is a pseudo-equilibrium condition. It is not a strict thermodynamic equilibrium condition.