You’ll first need to determine what format your existing events are in, and what format the new events need to be in. Most plugins save events as a custom post type, so you’ll usually find data on them in both wp_posts and wp_postmeta in the database.
Once you determine the format they’re in and how that needs to change, you can either run queries directly in phpMyAdmin, or set up PHP to loop through and update them for you. If you run queries directly, and you’re certain that both plugins store the events in the Posts table, you’ll likely need to start with something like
UPDATE wp_posts SET post_type="event-organizer" WHERE post_type="event"
(If you are using a different database prefix, you’ll need to change wp_posts to yourcustomprefix_posts.)
But, you’ll also likely need to change some of the postmeta. It all just depends on the original format and the target format. You may want to contact the plugin authors or see if they have documentation, if you’re not comfortable browsing through the database to determine where everything is stored and what format it’s in.
Deciding whether to set up the new event type from scratch, or convert the old events, will mostly come down to how comfortable you are determining the formats and running queries or code, and how many events you have – even if you’re comfortable with code it may simply be faster to copy them over by hand.
Also keep in mind that your URLs may change with the new plugin, and if so, you’ll want to set up redirects so anyone with the old links (including search engines) can seamlessly hop to the new URLs.