There is no doubt the two words are close in meaning in many circumstances, so there is some overlap. I have long tutored for college and other entrance exams, on which there is often a premium placed on subtle differences in meaning, and perhaps this might help:
If a person looks dubious, the primary meaning is doubtful (you can see the same root in the words), for example, "I told him the watch I was selling was made of solid gold but he gave me a dubious look." Meaning, he had some doubts that what I was saying was true. A secondary meaning, often when applied to a thing or a situation, means of doubtful quality or questionable, for example, "He made the dubious assertion that the watch was made of solid gold."
Suspicious is less about a sense of doubt and more about a sense of the (potentially evil) unknown. For example, "My neighbor was extremely paranoid and gave everyone who walked past his house a suspicious look." Meaning, he suspects some evil or untrustworthy motives. When applied to a thing, it has a similar sense, as in "Whenever the police find a suspicious package on the subway they must evacuate the entire train until they can verify that there is no danger."
Reasonable people may differ, and there are slangy ways to use either word, but I think the core issue comes down to doubt (dubious) versus the general threat of the undesirable (suspicious). Hope it helps!