Catching two error messages/Throw into table
You can enhance error handling with Extended Events to overcome the TRY...CATCH shortcoming you are experiencing.  The design pattern involves these steps:


Create and start an Extended Events Session: it will capture sqlserver.error_reported events, filtered primarily by SPID.
Execute a statement in a TRY block.
Within (or after) the CATCH block, read the XEvents session data.
Use the available data to respond to the error(s) as appropriate.
Stop and drop the XEvents session.


Here's an example that can be run manually in a single batch from SSMS. Just make sure to replace all occurrences of "2016" in the script with your SPID.

CREATE EVENT SESSION [Error Handling Session(2016)]
ON SERVER 
ADD EVENT sqlserver.error_reported
(
    ACTION(
        sqlserver.session_id,
        sqlserver.sql_text
    )
    WHERE [package0].[not_equal_unicode_string]([message],N'''''') 
    AND [severity]&gt;(10) 
    AND [sqlserver].[session_id]=(2016)
) 
ADD TARGET package0.ring_buffer
WITH (
    --ALLOW_SINGLE_EVENT_LOSS
    --NO_EVENT_LOSS
    EVENT_RETENTION_MODE=ALLOW_SINGLE_EVENT_LOSS,

    MAX_MEMORY=4096 KB,
    MAX_DISPATCH_LATENCY=1 SECONDS,
    MAX_EVENT_SIZE=0 KB,
    MEMORY_PARTITION_MODE=NONE,
    TRACK_CAUSALITY=ON,
    STARTUP_STATE=OFF
);

ALTER EVENT SESSION [Error Handling Session(2016)] ON SERVER STATE=START;

BEGIN TRY
    BACKUP DATABASE master
    TO DISK = 'master.diff.bak'
    WITH DIFFERENTIAL;
END TRY
BEGIN CATCH
    DECLARE @XEData XML

    SELECT @XEData = CAST(xet.target_data AS XML)
    FROM sys.dm_xe_session_targets AS xet
    JOIN sys.dm_xe_sessions AS xe
        ON (xe.address = xet.event_session_address)
    WHERE xe.name = 'Error Handling Session(2016)';

    /*
    Check value of "totalEventsProcessed" to ensure events have been 
    dispatched to event session target (ring_buffer).
    If no events have been processed, delay for a period of MAX_DISPATCH_LATENCY +1 (in seconds).
    */
    IF @XEData.value('(/RingBufferTarget/@totalEventsProcessed)[1]', 'INT') = 0
    BEGIN
        WAITFOR DELAY '00:00:02';

        SELECT @XEData = CAST(xet.target_data AS XML)
        FROM sys.dm_xe_session_targets AS xet
        JOIN sys.dm_xe_sessions AS xe
            ON (xe.address = xet.event_session_address)
        WHERE xe.name = 'Error Handling Session(2016)';
    END

    --Comment/uncomment as desired to show the formatted XML session data.
    SELECT @XEData;

    --Shred the XML. Do whatever you want with this data: 
    -- log it to a table, create and send an email alert, etc.
    SELECT 
        x.c.value(N'(@name)[1]', N'NVARCHAR(MAX)') AS EventName,
        x.c.value(N'(@timestamp)[1]', N'datetime') AS EventTime,
        x.c.value(N'(data[@name="error_number"]/value)[1]', N'NVARCHAR(MAX)') AS ErrorNumber,
        x.c.value(N'(data[@name="severity"]/value)[1]', N'NVARCHAR(MAX)') AS Severity,
        x.c.value(N'(data[@name="state"]/value)[1]', N'NVARCHAR(MAX)') AS [State],
        x.c.value(N'(data[@name="user_defined"]/value)[1]', N'NVARCHAR(MAX)') AS UserDefined,
        x.c.value(N'(data[@name="category"]/text)[1]', N'NVARCHAR(MAX)') AS Category,
        x.c.value(N'(data[@name="destination"]/text)[1]', N'NVARCHAR(MAX)') AS Destination,
        x.c.value(N'(data[@name="is_intercepted"]/value)[1]', N'NVARCHAR(MAX)') AS IsIntercepted,
        x.c.value(N'(data[@name="message"]/value)[1]', N'NVARCHAR(MAX)') AS [Message],
        x.c.value(N'(action[@name="sql_text"]/value)[1]', N'NVARCHAR(MAX)') AS SqlText,
        x.c.value(N'(action[@name="session_id"]/value)[1]', N'NVARCHAR(MAX)') AS SessionId
    FROM @XEData.nodes('//RingBufferTarget/event') AS x(c)
    --WHERE x.c.value(N'(data[@name="destination"]/text)[1]', N'NVARCHAR(MAX)') = 'USER'

END CATCH

ALTER EVENT SESSION [Error Handling Session(2016)] ON SERVER STATE=STOP;
DROP EVENT SESSION [Error Handling Session(2016)] ON SERVER;
GO


After running the script, your SSMS output should look similar to this:


There's a lot of moving parts there. But you might be able to make it work for you. I wrote a few related blog posts about TRY...CATCH that may be helpful:

The Unfulfilled Promise of TRY...CATCH

Enhanced T-SQL Error Handling With Extended Events

Part 2: Enhanced T-SQL Error Handling With Extended Events