I'll take on this question from the usability and extensibility point of view. For example, if I'm planning to use a protocol to exchange data, say text messages between users, over the network between two applications my first choice would be HTTP.
Why would I choose HTTP instead of something simpler, for example raw TCP with some hand made header on top? Because application grow, and I know that in two months time I'll be extending this simple hand crafted header to add text encoding. In four months time I'll extend the header again to allow for chunked transfer. And in 3 years time I'll end with a horrible to maintain and document hand crafted header that is probably buggy as hell (including security implications).
If I choose to use HTTP, at first it will look bloated. But after 3 years I'll discover that 80% of the extensions I have added are conformant to some part of the HTTP protocol. So I need to document less things, and need to maintain less code myself.
And now back to OpenID 2.0 vs. OpenID Connect.
OpenID Connect is layered on top of OAuth 2.0, OpenID 2.0 isn't. For a long time OpenID was a concurrent system to OAuth, yet, they both have a slightly different philosophy of usage:
So yeah, you are looking for a solution in which that users from one place can access another place using the same credentials. You are after OpenID not OAuth.
Yet, it is highly likely that at some point in the future you will want to extend this interaction between the two websites and share not only credentials but also data between the two. If you implement OpenID 2.0 now, you will need to implement OAuth at that point and will have a cranky piece of software in which there are two different types of credentials floating around. And that is quite error prone.
If you go for OpenID Connect, then, when your users will want to share data, you will have the scaffolding already built. All you will need to do is to add extra processing of the valet key and your application is clean.
Of course, it is only likely that at some point in the future the interaction between the websites will need to increase. And to evaluate this likeness is beyond the information that can be written in a discussion of the comparison of the advantages and disadvantages of the two protocols. Yet, I'd certainly prefer having an ace in the hole for when such a situation arrives, rather than risk ending with an ugly and error prone extension.