Both are literally different.
The first query does the filtering of table t2 before the joining of tables take place. So the results will then be join on table t1 resulting all the records of t1 will be shown on the list.
The second one filters from the total result after the joining the tables is done.
Here's an example
Table1
ID   Name
1    Stack
2    Over 
3    Flow
Table2
T1_ID   Score
1       10
2       20
3       30
In your first query, it looks like this,
SELECT  a.*, b.Score
FROM    Table1 a
        LEFT JOIN Table2 b
           ON a.ID = b.T1_ID AND
              b.Score >= 20
What it does is before joining the tables, the records of table2 are filtered first by the score. So the only records that will be joined on table1 are
T1_ID   Score
2       20
3       30
because the Score of T1_ID is only 10. The result of the query is 
ID   Name    Score
1    Stack   NULL
2    Over    20
3    Flow    30
While the second query is different.
SELECT  a.*, b.Score
FROM    Table1 a
        LEFT JOIN Table2 b
           ON a.ID = b.T1_ID
WHERE   b.Score >= 20
It joins the records first whether it has a matching record on the other table or not. So the result will be
ID   Name    Score
1    Stack   10
2    Over    20
3    Flow    30
and the filtering takes place b.Score >= 20. So the final result will be
ID   Name    Score
2    Over    20
3    Flow    30