What does the following sentence mean?
Spiders are not strictly insects.
Does it have the same meaning as
Strictly speaking, spiders are not insects.
It means that spiders, even though they are considered by many people to be insects, they are not technically insects. Strictly is used like shorthand for:
Spiders are not insects by the strictest definition.
Another word you could use instead of strictly is technically. This has the same definition.
Spiders are not technically insects.
With regards to the second part of your question, both sentences are effectively the same in meaning.