ok, most common cause is outlook is removing emails from server after download
Click Tools, then Accounts Settings from the menu bar:
Select Your account to change.
Highlight the account you want to change, then click Change to open the settings window.
The "Change E-mail Account" window opens.
Open the Advanced settings screen.
Click the "More Settings" button, then choose the "Advanced" tab.
In the "Advanced" tab, check the Leave a copy of messages on the server checkbox.
Save this, then send a new email and it should now appear on both computer and the iPhone
Let me know the results please