Scenario 1: Bob points a gun at Joe's head, says "I'll kill you", pulls the trigger, and misses Joe by 1 inch. Joe is fine physically.
Scenario 2: Bob points a gun at Joe's head, says "I'll kill you", pulls the trigger, and shoots him as intended. Doctors just barely save Joe from death.
Are these both attempted murder? Is this the same offense with the same penalty?
In the United States, this would be attempted murder in both cases, though in the case of actual physical harm, the prosecutor could charge related offenses such as battery, which is "an intentional offensive or harmful touching of another person that is done without his or her consent."
Note that though most attempted murders would likely be state crimes, I'm going to answer the rest of this from the perspective of a federal prosecution for attempted murder. The result would likely be similar for states, though the exact mechanism would be different.
Sentencing for people convicted of serious federal crimes is guided by the United States Federal Sentencing Guidelines. An "offense level" is determined, then combined with the offender's criminal history and checked against the sentencing table to determine a suggested sentencing range for the judge.
Specifically, for Assault with Intent to Commit Murder; Attempted Murder, §2A2.1. states that:
(a) Base Offense Level:
(1) 33, if the object of the offense would have constituted first degree murder; or
(2) 27, otherwise.
(b) Specific Offense Characteristics
(1) If (A) the victim sustained permanent or life-threatening bodily injury, increase by 4 levels; (B) the victim sustained serious bodily injury, increase by 2 levels; or (C) the degree of injury is between that specified in subdivisions (A) and (B), increase by 3 levels.
(2) If the offense involved the offer or the receipt of anything of pecuniary value for undertaking the murder, increase by 4 levels.
So in this case, it sounds like Joe suffered a life-threatening injury in the scenario where he got shot. Let's assume that this is Bob's first offense, and that it would have constituted first degree murder.
If Bob missed, he'd be looking at 135-168 months (11.25-14 years) in prison. If he shot Joe successfully, he'd be instead looking at 210-262 months (17.5-21.83 years) in prison.
Both scenarios have the same mens rea (intent to kill) and actus reus (firing the gun) so in england-and-wales they are both attempted murder - plus the potential for host of other off topic offences.
The difference will come at sentencing if Bob is found guilty. The Sentencing Council's guidelines provide a sliding scale for judges depending on the...
...seriousness of the offence, harm caused to the victim, the offender’s level of blame, their criminal record, their personal circumstances and whether they have pleaded guilty.
It is common to file both aggravated assault charges and attempted murder charges in cases where an attempted murder results in serious injuries. The attempted murder charge typically requires a stronger showing of ill intent so it is harder to prove and typically carries a longer sentence (subject to the discretion of the sentencing judge who typically has more freedom at the state level than in the federal courts).
If there is not injury, one has to go forward only on the attempted murder charges and not the aggravated assault charges, which can be harder to prove but authorizes the same maximum sentence if there is a conviction. Sometimes there would be a lesser included charge of menacing (i.e. trying to scare someone with a firearm unlawfully) in an attempted murder charge which would carry a felony sentence if there was no attempted murder conviction, but a less serious potential sentence than an aggravated assault conviction.
