Mozart spent money like a sailor on leave. Yes, he was incredibly famous, and performed for most of the crowned heads of Europe during his grand tour as a child, and composed prolifically for Emperors and nobility alike, he still didn't make enough to keep up with his lifestyle.

If you read some of Mozart's letters, particularly his letters to his father, he frequently laments about not having enough money.

"At present I have only one pupil. I could have several if I were to lower my fee; but as soon as one does that one loses credit. My price is twelve lessons for six ducats, and I make it understood besides that I give the lessons as a favor. I would rather have three pupils who pay well than six who pay ill. I am writing this to you to prevent you from thinking that it is selfishness which prevents me from sending you more than thirty ducats."

"Resent anything and at once you receive smaller pay. Besides all this the Emperor is a skinflint. If the Emperor wants me he ought to pay for me; the mere honor of being in his employ is not enough. If the Emperor were to offer me 1,000 florins and a count 2,000, I should present my compliments to the Emperor and go to the count, assuming a guarantee, of course."

Mozart had a patronage with the Prince Archbishop of Salzburg, but was paid very little, and the Prince Archbishop attempted, sometimes successfully, to prevent Mozart from performing elsewhere. After leaving Salzburg, Mozart received offers to be an organist, but refused them because he wanted to compose (a trade that's as difficult to make money in then as it is now). While he did compose for the Emperor, he received intermittent commissions, and not an actual salary for them, same for all his commissions. A commission is a lump sum, rather than an hourly rate, and you wouldn't receive residuals from performances the way that you do today.

Now pair that with a somewhat extravagant lifestyle, buying expensive clothes and living beyond his means, which, again, he mentions in his letters:

"I could not go about Vienna looking like a tramp, particularly just at this time. My linen was pitiable; no servant here has shirts of such coarse stuff as mine, and that certainly is a frightful thing for a man. Consequently there were again expenditures. I had only one pupil ; she suspended her lessons for three weeks, and I was again the loser. One must not throw one's self away here, that is a first principle, or one is ruined forever. The most audacious man wins the day."

And you get an idea of the problem. While there were times Mozart received sizeable commissions, there were times when he didn't, and lived off of money from his father, or gifts from friends. At the time of his death, there was virtually no money, so they couldn't afford any kind of lavish funeral service.