Is there any difference at all here?
(I read one of them in a Wikipedia article and the other one in my textbook)
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標高 refers specifically to the height above sea level of any point in Japan. This height is measured with the average level of the water in Tokyo Bay as zero. There is a stone marker in Nagata-cho, Tokyo, which is 24.39 metres above this level and is used as the national reference point.
海抜 means the height above the average level of the adjacent sea of any point on the planet.
First of all, Graham Healey's answer is accurate and very good.
I am a Japanese. When I hear the two words at the same time I think like:
Apart from the scientific accuracy, usually we don't think about the difference between them and I think they have the same meaning.
Both words have academic nuance. As for the altitude of a certain mountain, we usually say 「山｛やま｝の高｛たか｝さ」 instead of saying neither 「山｛やま｝の標高｛ひょうこう｝」 nor 「山｛やま｝の海抜｛かいばつ｝」.
To digress a little, I realized that we rarely say 「山｛やま｝の海抜｛かいばつ｝」 in comparison with 「山｛やま｝の標高｛ひょうこう｝」.
The usage of 「富士山｛ふじさん｝の高｛たか｝さ」 and 「富士山｛ふじさん｝の標高｛ひょうこう｝」 are natural, but 「富士山｛ふじさん｝の海抜｛かいばつ｝」 sounds queer though it might be grammatically correct.
I thought about why we don't say 「富士山｛ふじさん｝の海抜｛かいばつ｝」, then I found out that the word 海抜｛かいばつ｝ is not commonly used independently but used accompanied with the actual height in a set phrase such as 「富士山｛ふじさん｝は海抜｛かいばつ｝３７７６メートルです」, on the other hand the word 標高｛ひょうこう｝ is used independently and also in a set phrase.
If I arrange them you can say as:
And, You can't say as:
So, I image from the word 標高｛ひょうこう｝ that it points the height of an object which is in the obvious scenery where I am.
And I image from the word 海抜｛かいばつ｝ that I am looking at the object placed in a panoramic scenery including the sea.