This is not the best example. Since 'entering' is not something a person can just partially do (they either enter or they don't, like being pregnant), the contrast is not very strong. As such, whatever the police saw, they must have inferred that the process of the man entering the bank ran to its completion.
A better example would be
The police saw the man eating the hot dog.
The police saw the man eat the hot dog.
Both of these sentences report a fact in a very general way. In either case, whether the police saw the man finish eating the hot dog is an open question.
Note, however, that only the first of the following two is acceptable:
The police saw the man eat the hot dog and then leave.
The police saw the man eating the hot dog and then leave. (wrong)
You could say, however,
The police saw the man eating the hot dog finish [it] and then leave.
Also, there is one special way in which the original sentences are different:
The police observed the man entering the bank
can answer the question
Who(m) did the police observe?
but the other sentence cannot:
The police observed the man enter the bank* (wrong).
This is because "entering the bank" can be the complement to the verb "observed" (compare "I love entering the bank!") OR a modifier to the noun "man" (compare "People entering banks are usually in a bad mood.")
By contrast, "enter the bank" can ONLY be the complement to the verb "observed", since "the man enter the bank" makes no sense on its own as a noun phrase (compare "the man entering the bank")