Yes, all of the three mean different in some way but they have different usages to them.
This is the only word that will really work alone; without specifying what difference in respect to what is meant.
Der andere Hund. (The other/different dog)
Der Hund ist anders. (The dog is different)
Note that jemand ist anders nowadays is also used euphemisticly to signify a non-standard mental condition, so be careful when using that construction with humans.
Der eine and der andere can also contrast to mean one and the other.
Das ist Wuffi, der eine Hund, und das ist Kläffi, der andere Hund.
As mentioned, this contrasts two things. It kind of implies that there is a certain attribute that differs between the two things (coming from the verb unterscheiden, to differ) and somehow more emphasising this minute difference.
Wuffi und Kläffi sind unterschiedlich. Sie unterscheiden sich an den Ohren: Wuffi hat Stehohren, Kläffi hat Schlappohren.
(Wuffi and Kläffi are different. They differ at their ears: Wuffi’s ears are standing upright, Kläffis ears are floppy.)
This just means plain different, without specifying what the differences are in any way.
Natürlich sind Kläffi und Törti verschieden! Es sind schließlich verschiedene Rassen!
(Of course Kläffi and Törti are different! They represent different races!)
In Plural usage
Die anderen Hunde (wohnen draußen im Zwinger)
This means the other dogs, again contrasting to dogs you might have already seen.
Die verschiedenen Hunde, die hier wohnen, …
This means all of the different dogs here.
Die unterschiedlichen Hunde, …
This means that there are dogs that are different from each other while there are other dogs around that might all look the same … so a dalmatian and a Rhodesian ridgeback contrasting to five basset hounds or something.
I’m trying to think of how well one can replace either word with another. I don’t think anders can be replaced by either unterschiedlich or verschieden and vice-versa; that should usually change the meaning (from different to other). On the other hand, in some occasions verschieden and unterschiedlich can be exchanged without damaging the meaning too much. But I would recommend against it, just in case.
Finally, just to put it all into a nice example that doesn’t include dogs (although I don’t know why I would want to exclude dogs):
Ich habe Karten ausgeteilt. Du müsstest jetzt fünf verschiedene Karten auf der Hand haben. Wenn du fünf unterschiedliche Karten auf der Hand hast, ist dein Zug vorbei, und ich bin dran. Zwei gleiche Karten kannst du abwerfen, dafür bekommst du zwei andere Karten.
I dealt out cards. You should have five different (non-identical in the strictest sense) cards on your hand. If you have five different (in suit and/or rank; would need specification) cards on your hand, your turn is over and it is my turn. You can discard two same cards (sounds like suit and rank, but could be specified to something else); for those you will get two new (different?) cards.