I assume you know already how spring tides and leap tides occur and you only want to know where they got their name from.

Well, this is a tough question. As you know, the sun and moon each raise tidal bulges on the Earth's oceans. When the sun and moon are on the same side of the Earth (new moon) or the opposite side (full moon), their bulges add together to make larger tides than usual: this is called a "spring tide" . When the sun and moon are 90 degrees apart (first quarter or last quarter moon), the bulges interfere and cancel each other: this creates the unusually small "neap tide".

Now, especially high tides are called spring tides, but they have nothing to do with the season and actually occur twice a month. I think the name comes from the German word "Springen" meaning "to leap". Spring tides also mean lower low water. The opposite to a spring tide is a neap tide.

The derivation of neap tides, however, seems simpler. The Shorter Oxford dictionary gives the derivation of 'neap' from the Old English (OE) word 'nep', to become lower.

There is an article on the WWW that tries to explain the roots of the words "spring" and "neap". The author basically comes to the same conclusion although she believes "spring" comes from the Anglo-Saxon word springan (to bulge). But in the Middle Ages the German language and the Anglo Saxon were very close.

Neap tides and spring tides are very different.

Spring tides are very high and very low tides (that occur at the same time) whereas neap tides are those tides that are sort of the 'mid tides', not very high and not very low.

neap tide
A tide that occurs when the difference between high and low tide is least; the lowest level of high tide. Neap tide comes twice a month, in the first and third quarters of the moon.

spring tide
The exceptionally high and low tides that occur at the time of the new moon or the full moon when the sun, moon, and earth are approximately aligned.