All hosting companies offer you a choice in webmail programs. You can pick whichever one you are most familiar with, or the one you learn to like over time.
You can switch webmail programs as much as you like, and you will still see all your emails.
Horde is a popular choice, but does require some configuration to make it behave the way users expect. (For instance, Horde does not use a trash can for your deleted emails.)
SquirrelMail is highly recommended as the webmail program you should use. This software behaves the way users expect. (For example, it sends deleted mail to a trash folder.) Also, Squirrel Mail has low amounts of graphics, which means faster page loads. We also find that SquirrelMail is more reliable. If your other webmail programs are not working, try SquirrelMail.
RoundCube is another popular choice. Again, just pick the program you like.
Don’t forget, you can change webmail programs anytime, and you will still see the same emails.