What is the difference between sequence alignment and sequence assembly?
Sequence alignment
It is done for checking sequence similarity between two or more different sequences. This will give information about how two sequences are different, what is their evolutionary relationship, which residues are conserved etc. Take a look at following sequence alignment between different sequences. (Image courtesy: Wikimedia Commons)

You can see conversation of different amino acid residues in variety of organisms like human, mouse, rat etc.
Sequence Assembly
It is done for creating long consensus sequence from short fragments of same sequence. Generally you would do sequence assembly after sequencing some genome or large DNA fragment. Take a look at following representative image of sequence assembly (Image courtesy: Wikimedia Commons)

Here black sequence is your original sequence which you wanted to sequence. However because of technical difficulties you can't sequence all at once. Hence you fragment and sequence different fragments 9which are shown in red, blue and green). Looking at overlapping region between different sequenced fragments you calculate consensus (shown in Cyan). Here you are ideally sequencing only one large sequence and assembling it.