In TensorFlow, what is the difference between Session.run() and Tensor.eval()?

The FAQ session on tensor flow has an answer to exactly the same question. I will just go ahead and leave it here:

If t is a Tensor object, t.eval() is shorthand for sess.run(t) (where sess is the current default session. The two following snippets of code are equivalent:

sess = tf.Session()
c = tf.constant(5.0)
print sess.run(c)

c = tf.constant(5.0)
with tf.Session():
  print c.eval()

In the second example, the session acts as a context manager, which has the effect of installing it as the default session for the lifetime of the with block. The context manager approach can lead to more concise code for simple use cases (like unit tests); if your code deals with multiple graphs and sessions, it may be more straightforward to explicit calls to Session.run().

I’d recommend that you at least skim throughout the whole FAQ, as it might clarify a lot of things.

eval() can not handle the list object

tf.reset_default_graph()

a = tf.Variable(0.2, name="a")
b = tf.Variable(0.3, name="b")
z = tf.constant(0.0, name="z0")
for i in range(100):
    z = a * tf.cos(z + i) + z * tf.sin(b - i)
grad = tf.gradients(z, [a, b])

init = tf.global_variables_initializer()

with tf.Session() as sess:
    init.run()
    print("z:", z.eval())
    print("grad", grad.eval())

but Session.run() can

print("grad", sess.run(grad))

correct me if I am wrong

In tensorflow you create graphs and pass values to that graph. Graph does all the hardwork and generate the output based on the configuration that you have made in the graph.
Now When you pass values to the graph then first you need to create a tensorflow session.

tf.Session()

Once session is initialized then you are supposed to use that session because all the variables and settings are now part of the session. So, there are two ways to pass external values to the graph so that graph accepts them. One is to call the .run() while you are using the session being executed.

Other way which is basically a shortcut to this is to use .eval(). I said shortcut because the full form of .eval() is

tf.get_default_session().run(values)

You can check that yourself.
At the place of values.eval() run tf.get_default_session().run(values). You must get the same behavior.

what eval is doing is using the default session and then executing run().

The most important thing to remember:

The only way to get a constant, variable (any result) from TenorFlow is the session.

Knowing this everything else is easy:

Both tf.Session.run() and tf.Tensor.eval() get results from the session where tf.Tensor.eval() is a shortcut for calling tf.get_default_session().run(t)

I would also outline the method tf.Operation.run() as in here:

After the graph has been launched in a session, an Operation can be executed by passing it to tf.Session.run(). op.run() is a shortcut for calling tf.get_default_session().run(op).

Tensorflow 2.x Compatible Answer: Converting mrry’s code to Tensorflow 2.x (>= 2.0) for the benefit of the community.

!pip install tensorflow==2.1
import tensorflow as tf

tf.compat.v1.disable_eager_execution()    

t = tf.constant(42.0)
sess = tf.compat.v1.Session()
with sess.as_default():   # or `with sess:` to close on exit
    assert sess is tf.compat.v1.get_default_session()
    assert t.eval() == sess.run(t)

#The most important difference is that you can use sess.run() to fetch the values of many tensors in the same step:

t = tf.constant(42.0)
u = tf.constant(37.0)
tu = tf.multiply(t, u)
ut = tf.multiply(u, t)
with sess.as_default():
   tu.eval()  # runs one step
   ut.eval()  # runs one step
   sess.run([tu, ut])  # evaluates both tensors in a single step