What's the difference between a bidirectional LSTM and an LSTM?

LSTM in its core, preserves information from inputs that has already passed through it using the hidden state.

Unidirectional LSTM only preserves information of the past because the only inputs it has seen are from the past.

Using bidirectional will run your inputs in two ways, one from past to future and one from future to past and what differs this approach from unidirectional is that in the LSTM that runs backwards you preserve information from the future and using the two hidden states combined you are able in any point in time to preserve information from both past and future.

What they are suited for is a very complicated question but BiLSTMs show very good results as they can understand context better, I will try to explain through an example.

Lets say we try to predict the next word in a sentence, on a high level what a unidirectional LSTM will see is

The boys went to ....

And will try to predict the next word only by this context, with bidirectional LSTM you will be able to see information further down the road for example

Forward LSTM:

The boys went to ...

Backward LSTM:

... and then they got out of the pool

You can see that using the information from the future it could be easier for the network to understand what the next word is.

Adding to Bluesummer's answer, here is how you would implement Bidirectional LSTM from scratch without calling BiLSTM module. This might better contrast the difference between a uni-directional and bi-directional LSTMs. As you see, we merge two LSTMs to create a bidirectional LSTM.

You can merge outputs of the forward and backward LSTMs by using either {'sum', 'mul', 'concat', 'ave'}.

left = Sequential()
left.add(LSTM(output_dim=hidden_units, init='uniform', inner_init='uniform',
               forget_bias_init='one', return_sequences=True, activation='tanh',
               inner_activation='sigmoid', input_shape=(99, 13)))
right = Sequential()
right.add(LSTM(output_dim=hidden_units, init='uniform', inner_init='uniform',
               forget_bias_init='one', return_sequences=True, activation='tanh',
               inner_activation='sigmoid', input_shape=(99, 13), go_backwards=True))

model = Sequential()
model.add(Merge([left, right], mode='sum'))

model.add(TimeDistributedDense(nb_classes))
model.add(Activation('softmax'))

sgd = SGD(lr=0.1, decay=1e-5, momentum=0.9, nesterov=True)
model.compile(loss='categorical_crossentropy', optimizer=sgd)
print("Train...")
model.fit([X_train, X_train], Y_train, batch_size=1, nb_epoch=nb_epoches, validation_data=([X_test, X_test], Y_test), verbose=1, show_accuracy=True)