What is the difference between Sam Bankman Fried's "FTX" and any other bulge bracket bank?

Probably a bait post since it's asking such incredibly silly questions but I'll bite anyway.

1. Why was Sam Bankman Fried's company "FTX" relocation of customer funds to separate financial ventures considered illegal but when the big bulge bracket banks do it, its legal? The big bulge bracket banks like Chase use money from customer deposits/savings to lend out in the form of Mortgages and also use customer-money to support their own IB/PE/HF/Etc. 

Because Chase and other banks are regulated entities with licenses to operate as banks whereas FTX was neither regulated nor licensed. Banks are only allowed to use customer funds for specific types of loans/investments that are considered relatively low risk and as part of that are in some cases required to provide some sort of compensation to their depositors through either services or interest.

FTX was supposed to operate as an exchange which is permitted only to HOLD customer funds and enable trade execution. They explicitly stated throughout marketing, compliance, and T&C materials that they would NOT touch customer funds under any circumstances, which they then turned around and did anyway which is textbook FRAUD. It's incredibly obvious that they knew what they were doing was illegal because they created a backdoor in their own platform to access these customer funds without anyone else's knowledge and would then modify the entries of their own central general ledger to show that the customer funds were untouched whilst being siphoned off to gamble with in his market-making arm Alameda Research. That firm then proceeded to lose most of said customer funds taking directional bets both directly and through derivatives on the price of Bitcoin and other cryptocurrencies, risky activity that falls well outside the regulatory allowances of any bank.

To top it off, SBF took these customer deposits and illegally used them for marketing expenses e.g. buying stadium naming rights, Superbowl commercials, hire influencers, etc. as well as to lobby sitting and prospective members of the US Congress and Senate with ties to the various financial oversight committees. Put it simply, he stole people's money, lied about it, and proceeded to use it to try and entice more victims to join in + bribe politicians to the tune of 100s of millions of dollars to try and cover his ass. 

2. Why did the media make it a big deal when Sam Bankman Fried bought a $30 million mansion when he was worth $26 billion? $30M is so small compared to $26 billion

He was never worth $26 billion and was never "liquid" in that even $1b of that was probably never available to him outside of maybe loans against his equity. That was paper net worth based on the valuation at the time of the last fundraise. And there's no way to determine (though it's highly likely at this point) that he bought said $30m mansion using stolen customer funds. Makes sense seeing as how he was basically using customer deposits as a piggy bank for everything else.  

3. If Sam Bankman Fried's "FTX" is based in the Bahamas, and the "fraud" occurred there, how does the USA have jurisdiction to punish Sam Bankman Fried if this was done in the Bahamas?

He is a US citizen, committed securities fraud in the US by dealing with US-based investors, had a US operating arm that was taking on deposits which also were stolen/misused, and was bribing US politicians across multiple election cycles.

4. Why didn't the government bail out "FTX" in the same way they bailed out the banks that were "too big to fail"?

Silly question. He's not a bank, obviously. FTX's core business was not in the US. He was neither too big to fail nor tied deeply enough into the financial system that his collapse would effect the greater economy. He was just operating an elaborate ponzi scheme in the Bahamas. 

5. If Sam Bankman Fried was worth $26 billion, why didn't he just use his own money to fund his Alameda Research financial ventures(Along with leverage/legally borrowed capital)?

Refer to the above, he was never actually worth $26 billion. That's not how net worth works.

In conclusion:

The only thing SBF deserves is to hang out in the Bahamas or sit in a shockingly uncomfortable chair in the US for the rest of his life.