First of all, suicide is definitely not allowed in Islam:
The Prophet (ﷺ) said, "Whoever swears by a religion other than Islam, is, as he says; and whoever commits suicide with something, will be punished with the same thing in the (Hell) Fire; and cursing a believer is like murdering him; and whoever accuses a believer of disbelief, then it is as if he had killed him." (Sahih al-Bukari)
It is a grave sin that will lead to Hellfire. The difference between suicide and martyrdom is that in martyrdom there is hope that you will survive. For example, running towards to the enemy army without a weapon to die is suicide, while running towards them with a weapon in a well planned attack where your intention is to win and live then dying is martyrdom. In the first one, you are basically giving yourself to death, and in the second, there is a chance although small that you will live and win the fight.
So, when a suicide bomber goes to a marketplace or something with a bomb strapped to him, where is the hope of life? He is himself pressing the button to blow himself up. Taking his own life, which is suicide. Not even accounting for the fact that killing innocent people is haram, you can see that suicide bombing or any other type of suicide is definitely not allowed by Islam.
Furthermore killing innocent people, not involved in the battle or fight, is forbidden:
When we were with the Messenger of Allah (ﷺ) on an expedition, he saw some people collected together over something and sent a man and said: See, what are these people collected around? He then came and said: They are round a woman who has been killed. He said: This is not one with whom fighting should have taken place. Khalid ibn al-Walid was in charge of the van; so he sent a man and said: Tell Khalid not to kill a woman or a hired worker. (Sunan Abi Dawud)
“A woman was found slain in one of the battles of the Apostle of Allaah(ﷺ). The Apostle of Allaah(ﷺ) forbade to kill women and children." (Sunan Abi Dawud)
Suicide bombers deliberately target innocent people, women, and children (as can be seen in the case of 9/11) which is not allowed in Islam.
In conclusion, martyrdom is when a person dies fighting for a cause with a possibility of living, and he is fighting enemies that are not innocent women, children, or workers (i.e. only fighting soldiers). Suicide bombing breaks all of those rules.