Betrothal and engagement have very different connotations.
An engagement is a more modern term and a statement of intent to marry, often having a date already established, hence the saying among older generations that an engagement isn't one until there is a ring and a date set.
Betrothal dates from before marriage was a purely civil matter. Today, a marriage is not valid until a marriage license is issued and signed by the bride and groom with witnesses and the issuing official having signed as well in many countries. The prospective couple can be wed in a church, but it won't be recognized as valid until the marriage license is completed and submitted.
In many traditions, a betrothal essentially means that the couple is married in all but name, and was common until recently due to marriages often being held off for economic reasons as a husband was expected to be economically self-sufficient before marriage. This was especially true if he is the eldest (surviving) son and thus has to be able to take care of his parents in their old age. During a betrothal, the bride and groom still live with their respective families until the wedding, in which they will then live in the same house. There have been exceptions to this rule.
There is also the political aspect of betrothal, as arranged marriages were more common in the past for the purposes of furthering political and economic alliances, with the bride and groom often having years to get to know each other before the marriage.
In both cases, a betrothal was (and in some cultures, still is) as legally binding as a marriage. If either party sleeps with someone else other than their betrothed, it was, and still is in some cultures, considered adultery in the eyes of the law. Furthermore, a betrothal could only be ended through a divorce, unfaithfulness on the part of one or both of the parties.
Also, any children born during a betrothal were automatically considered legitimate without the father having to publicly acknowledge the child as his (with significant stigma if the father claims the child is not his but it can be proven that the mother had remained faithful) and is in line to inherit even if the father dies before the marriage.