The big difference is perfectly explained here.
Basically, lightweight tags are just pointers to specific commits. No further information is saved; on the other hand, annotated tags are regular objects, which have an author and a date and can be referred because they have their own SHA key.
If knowing who tagged what and when is relevant for you, then use annotated tags. If you just want to tag a specific point in your development, no matter who and when did that, then lightweight tags are good enough.
Normally you'd go for annotated tags, but it is really up to the Git master of the project.