Which is better: many join conditions or many where conditions?
If we consider that you use INNER JOIN instead of LEFT JOIN(which appears to be your intent), these two queries are functionally equivalent.  Query optimizers will review and evaluate criteria in your WHERE clause and your FROM clause and consider all of these factors when building query plans in order to reach the most efficient execution plan.  If we do an EXPLAIN on both statements, we get the same result:

Query 1:

EXPLAIN
SELECT 
  tableA.ColA
  ,tableA.ColB
  ,tableA.ColC
  ,tableA.ColD
  ,tableA.ColE
FROM tableA
  JOIN tableB ON tableA.ColA=tableB.ColA
WHERE 
  tableA.ColB=tableB.ColB 
  AND tableA.ColC=tableB.ColC 
  AND tableA.ColD=tableB.ColD  
  AND tableA.ColE=tableB.ColE


[Results]:

| ID | SELECT_TYPE |  TABLE | TYPE | POSSIBLE_KEYS |    KEY | KEY_LEN |    REF | ROWS |                          EXTRA |
------------------------------------------------------------------------------------------------------------------------
|  1 |      SIMPLE | tableA |  ALL |        (null) | (null) |  (null) | (null) |    1 |                                |
|  1 |      SIMPLE | tableB |  ALL |        (null) | (null) |  (null) | (null) |    1 | Using where; Using join buffer |


Query 2:

EXPLAIN
SELECT 
  tableA.ColA
  ,tableA.ColB
  ,tableA.ColC
  ,tableA.ColD
  ,tableA.ColE
FROM tableA
  JOIN tableB ON tableA.ColA=tableB.ColA
  AND tableA.ColB=tableB.ColB 
  AND tableA.ColC=tableB.ColC 
  AND tableA.ColD=tableB.ColD  
WHERE
  tableA.ColE=tableB.ColE


[Results]:

| ID | SELECT_TYPE |  TABLE | TYPE | POSSIBLE_KEYS |    KEY | KEY_LEN |    REF | ROWS |                          EXTRA |
------------------------------------------------------------------------------------------------------------------------
|  1 |      SIMPLE | tableA |  ALL |        (null) | (null) |  (null) | (null) |    1 |                                |
|  1 |      SIMPLE | tableB |  ALL |        (null) | (null) |  (null) | (null) |    1 | Using where; Using join buffer |


You can review the full details with the following links.  I also created a SQL 2008 example so that you can compare how the two engines work (which is the same):

MySQL query example

SQL 2008 query example (Make sure you 'View Execution Plan' for both results)