The private key itself does not actually change. What changes is its "export format". When the same 256-bit private key is exported in WIF format, if the "compressed public key" mode is selected, the encoding includes an extra byte marker 0x01. The exported string is then one character longer than the "uncompressed" mode, and the prefix changes from 5 to K or L. After a wallet imports this WIF, it uses this marker to decide whether to derive the address with a compressed public key or an uncompressed public key. The two public key serialization methods are different, so the calculated Bitcoin address is naturally different as well.
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Where the WIF length difference comes from
A Bitcoin private key is essentially a 256-bit number, represented in hexadecimal as 32 bytes or 64 characters. But the "private key" you export from a wallet is usually not this raw number. Instead, it is a WIF (Wallet Import Format) encoded string. WIF does several things: it adds a network prefix (mainnet uses 0x80), adds a checksum, and then encodes the result with Base58. The resulting string is shorter than the hexadecimal form and includes error-checking ability.
The key point is that if this private key will be used to generate a compressed public key, the WIF encoding adds an extra byte 0x01 at the end of the private key. This tells the wallet that the public key derived from this private key should be stored in compressed format. This extra byte changes the Base58 encoding result: an uncompressed WIF on mainnet starts with 5 and is usually 51 characters long; a compressed WIF starts with K or L and is 52 characters long.
So "different lengths" refers to these two WIF formats, not two different private keys.
Why the address also changes
A Bitcoin address is derived from a public key: the public key goes through SHA-256 and RIPEMD-160 to get a hash, and then a version byte and checksum are added. The issue is that the same private key can produce one public key point through elliptic curve multiplication, but this point has two serialization methods:
Uncompressed format: stores the full x coordinate and y coordinate of the point. The public key length is 65 bytes (04 + 32-byte x + 32-byte y).
Compressed format: stores only the x coordinate, plus one byte of 02 or 03 to indicate whether the y coordinate is even or odd. The public key length is 33 bytes.
These two different byte sequences go through the same hashing and encoding process, but the resulting addresses are completely different. One private key corresponds to two valid public keys and two valid addresses. When you import a WIF, the wallet checks whether the WIF has the 0x01 suffix. Based on that, it decides which public key format to use for deriving the address, so the final address shown is different.
What this means in practice
If an old wallet gives you a WIF starting with 5, and you import it into a newer wallet, the new wallet may treat it as an uncompressed private key and derive the old-format address starting with 1. If you previously sent coins to the address corresponding to the compressed format, you will not see those funds at the uncompressed address. The reverse is also true. The coins are not lost. The wallet is simply looking for the address that matches the public key mode indicated by the WIF you provided.
This is also why the WIF and WIF-compressed formats of the same private key cannot be used interchangeably: they carry different information and tell the wallet to do different things. If the address does not match after importing a private key, first check whether your WIF starts with 5 or with K/L, and then confirm whether the target wallet supports the public key mode you need.
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