They will be distinguishable, but they will still be bech32 addresses.
The introduction of Schnorr signatures requires a new type of output. Segwit was designed with such extensibility in mind, and it defines 17 version numbers. Currently, only version 0 is used. Segwit v0 outputs with a 20-byte hash are known as P2WPKH outputs, and v0 outputs with a 32-byte hash are known as P2WSH outputs.
A new version number can be introduced, for example v1, and given semantics through a softfork. Schnorr signatures is one of the changes being considered for a proposal.
However, bech32 addresses literally encode a version number plus a payload, which maps directly to the various versions of segwit outputs.
The version number in Bitcoin bech32 addresses is in the 4th character. For all v0 outputs, that version character is 'q'. For v1 outputs it will be a 'p'.
EDIT June 2022: Taproot was introduced with a slightly different address format than bech32, named bech32m, described in BIP350. This was done to address a shortcoming in the bech32 checksum algorithm, as well as prevent misinterpretation of taproot addresses as P2WSH (which would lead to loss of funds).
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