I can think of a few situations where 'suggest' cannot appropriately be replaced by 'advise', but I cannot think of any situations where 'advise' cannot be replaced by 'suggest'.
Here are two examples where 'suggest' seems to be the obvious choice:
Maddam, may I suggest the veal?
I suggest that we change the subject.
Also, don't forget that the words directly following 'suggest' and 'advise' aren't always the same, even when they ('suggest and 'advise') are interchangeable.
Here's an example:
I would suggest that you get a job.
I would advise you to get a job.