Both languages are fully dynamic and support multiple dispatch. But whereas Dylan opted to have both single and multiple dispatch, as well as dynamic and static dispatch, Julia only has one kind of dispatch: full dynamic multiple dispatch.
I'm not sure how Dylan passes variables and whether it separates reference types and values types, but Julia doesn't, instead opting to distinguish immutable versus mutable types. Immutables have many of the benefits of value types for the compiler, plus a few more. Mutable and immutable values have identical semantics other than that fact that you can't change an immutable value, which makes the programming model much simpler (all values have reference semantics), while giving the performance of value types for immutables since the compiler is free to pass them by value when it sees fit.
I believe Dylan allows you to inherit from concrete types by structural inheritance – i.e. tacking fields onto the fields of the parent type. in Julia all concrete types are final – you cannot inherit from them. I haven't found that tacking fields onto a structure is all that useful; sharing behavior is far more useful than sharing structure, and delegation is generally a better way to share structure.
Abstract types in Julia are essentially what OO researchers call traits: abstract types that you can program to. I don't think that Dylan has traits. These are crucial to much generic programming (the other kind of generic programming is writing generic code for parametric types; although these are actually the same thing in Julia).
Dylan supports multiple inheritance, while Julia does not. We might add multiple inheritance in the future – or not: so far it's not terrible to live without it, although there are a few places where it might be nice. In the meantime, duck typing will suffice, and once the standard library is complete and stable it will be much easier to assess if we really need multiple inheritance or not; if the standard library doesn't need it, I think the rest of the world can live without it.
The biggest philosophical differences seem to stem from:
1. Dylan was created in a time when OOP was new and hot and had to be included; Julia doesn't really care much about OOP – generic and functional programming are much more important.
2. Dylan tends to give you lots of options (e.g. single or multiple dispatch, static or dynamic), whereas Julia tends to give you a single option, but it's the most powerful one (only dynamic multiple dispatch). This keeps the language simpler and easier to learn and use, but no less powerful.