What is the difference between Model.train_on_batch from keras and Session.run([train_optimizer]) from tensorflow?

In the following Keras and Tensorflow implementations of the training of a neural network, how model.train_on_batch([x], [y]) in the keras implementation is different than sess.run([train_optimizer, cross_entropy, accuracy_op], feed_dict=feed_dict) in the Tensorflow implementation? In particular: how those two lines can lead to different computation in training?:

keras_version.py

input_x = Input(shape=input_shape, name="x")
c = Dense(num_classes, activation="softmax")(input_x)

model = Model([input_x], [c])
opt = Adam(lr)
model.compile(loss=['categorical_crossentropy'], optimizer=opt, metrics=['accuracy'])

nb_batchs = int(len(x_train)/batch_size)

for epoch in range(epochs):
    loss = 0.0
    acc = 0.0
    for batch in range(nb_batchs):
        x = x_train[batch*batch_size:(batch+1)*batch_size]
        y = y_train[batch*batch_size:(batch+1)*batch_size]

        loss_batch, acc_batch = model.train_on_batch([x], [y])

        loss += loss_batch
        acc += acc_batch
    print(epoch, loss / nb_batchs, acc/nb_batchs)

tensorflow_version.py

input_x = Input(shape=input_shape, name="x")
c = Dense(num_classes)(input_x)

input_y = tf.placeholder(tf.float32, shape=[None, num_classes], name="label")
cross_entropy = tf.reduce_mean(
    tf.nn.softmax_cross_entropy_with_logits_v2(labels=input_y, logits=c, name="xentropy"),
    name="xentropy_mean"
)
train_optimizer = tf.train.AdamOptimizer(learning_rate=lr).minimize(cross_entropy)
predictions = tf.argmax(c, 1)
correct_prediction = tf.equal(predictions, tf.argmax(input_y, 1))
accuracy_op = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))

nb_batchs = int(len(x_train)/batch_size)

init = tf.global_variables_initializer()
with tf.Session() as sess:
    sess.run(init)
    for epoch in range(epochs):
        loss = 0.0
        acc = 0.0

        for batch in range(nb_batchs):
            x = x_train[batch*batch_size:(batch+1)*batch_size]
            y = y_train[batch*batch_size:(batch+1)*batch_size]

            feed_dict = {input_x: x,
                         input_y: y}
            _, loss_batch, acc_batch = sess.run([train_optimizer, cross_entropy, accuracy_op], feed_dict=feed_dict)

            loss += loss_batch
            acc += acc_batch
        print(epoch, loss / nb_batchs, acc/nb_batchs)

Note: This question follows Same (?) model converges in Keras but not in Tensorflow , which have been considered too broad but in which I show exactly why I think those two statements are somehow different and lead to different computation.

from What is the difference between Model.train_on_batch from keras and Session.run([train_optimizer]) from tensorflow?