Compared to what?
The key to your question:
What are some disadvantages for using the "Greater than or Equal to" instruction in a rung to turn an output?
is, as Paul pointed out,
as opposed to what?
One possible answer to Paul's question is,
as opposed to serial timers.
Compare the following code:
YOURS
I:1/4 +---- TOM --+
---| |----+---| T4:2 |
| | PRE: 1500 |
| +-----------+
|
| O:0/4
+----------( )--
+---- GEQ --+ O:0/3
--| T4:2.ACC |------( )--
| 500 |
+-----------+
+---- GEQ --+ O:0/2
--| T4:2.ACC |------( )--
| 1000 |
+-----------+
Verses THIS
I:1/4 +---- TOM --+
---| |----+---| T4:4 |
| | PRE: 500 |
| +-----------+
| O:0/4
+----------( )--
T4:4.DN +---- TOM --+
----| |---+---| T4:3 |
| | PRE: 500 |
| +-----------+
|
| O:0/3
+----------( )--
T4:3.DN +---- TOM --+
----| |---+---| T4:2 |
| | PRE: 500 |
| +-----------+
|
| O:0/2
+----------( )--
NOW we can talk about advantages and disadvantages.
The first example (yours) uses just one timer, with various "Milestones" along the way. If we want to change delay for when O:0/3 comes on, from 5 seconds to 6 seconds, we just change the value in the GEQ, and only O:0/3 is affected. O:0/2 still comes on after 10 sec.
In the second method, if we want to change the delay for turing on O:0/3, we change the T4:4.PRE. But in doing so, we also affect when O:0/2 comes on. Now it comes on at 11 seconds after the button has been pressed, but delay BETWEEN the two remains the same.
Another difference. Timer presets are stored in registers. It is possoble to change those values without making changes to the program, using software or an HMI (Human Machine Interface - a specialized PC that only talks to/from PLCs and can't do anything else (like play Quake).
If you wanted to give the operator the abilty to change the timings, using timers allows that. In your example, the spaces are "hard-coded", and can't be changed without changing the program. (The GEQ can be modified to use registers instead of constants, giving it the same flexability as timers. But if you want to make those timings hard and fast, only the GEQ would allow that.
Or perhaps the answer to Paul's question is:
as opposed to using an EQU instruction.
In that case, the O:0/3 would only be on for the one scan where T4:2.ACC = 500 (if at all - depending on scan time, the ACC could equal 498 on one scan and 503 the next (if there was a 50 msec scantime). Thus it never EQUALS 500, and so O:0.3 never comes on. This type of mistake as bitten many a new PLC programmer.
The next time your professor (or anyone, for that matter) asks you to give the advantages or disadvantages of something, ask them "Compared to what"?
The reason your mind went blank is that you only had half of the problem.