They are two different protocols of authentication and they differ at the technical level.
From a distance, differences start when users initiate the authentication. With OpenID, a user login is usually an HTTP address of the resource which is responsible for the authentication. On the other hand, SAML is based on an explicit trust between your site and the identity provider so it’s rather uncommon to accept credentials from an unknown site.
OpenID identities are easy to get around the net. As a developer you could then just accept users coming from very different OpenID providers. On the other hand, a SAML provider usually has to be coded in advance and you federate your application with only selected identity providers. It is possible to narrow the list of accepted OpenID identity providers but I think this would be against the general OpenID concept.
With OpenID you accept identities coming from arbitrary servers. Someone claims to be http://someopenid.provider.com/john.smith . How you are going to match this with a user in your database? Somehow, for example by storing this information with a new account and recognizing this when user visits your site again. Note that any other information about the user (including his name or email) cannot be trusted!
On the other hand, if there’s an explicit trust between your application and the SAML Id Provider, you can get full information about the user including the name and email and this information can be trusted, just because of the trust relation. It means that you tend to believe that the Id Provider somehow validated all the information and you can trust it at the application level. If users come with SAML tokens issued by an unknown provider, your application just refuses the authentication.
OpenID Connect vs SAML
(section added 07-2017)
This answer dates 2011 and at that time OpenID stood for OpenID 2.0. Later on, somewhere at 2012, OAuth2.0 has been published and in 2014, OpenID Connect (a more detailed timeline here).
To anyone reading this nowadays – OpenID Connect is not the same OpenID the original answer refers to, rather it’s a set of extensions to OAuth2.0.
While this answer can shed some light from the conceptual viewpoint, a very concise version for someone coming with OAuth2.0 background is that OpenID Connect is in fact OAuth2.0 but it adds a standard way of querying the user info, after the access token is available.
Referring to the original question – what is the main difference between OpenID Connect (OAuth2.0) and SAML is how the trust relation is built between the application and the identity provider:
SAML builds the trust relation on a digital signature, SAML tokens issued by the identity provider are signed XMLs, the application validates the signature itself and the certificate it presents. The user information is included in a SAML token, among other information.
OpenID Connect (OAuth2.0) builds the trust relation on a direct HTTPs call from the application to the identity. The request contains the access token (obtained by the application during the protocol flow) and the response contains the information about the user.