Profit Maximization and Wealth Maximization are two objectives of Financial Management.
Financial Management takes cares for proper utilization of funds, such that it will increase company earnings.
Profit Maximization refers to the profit of the firm should be increased while in Wealth Maximization objective of a firm is to maximise its wealth and the value of its shares.
There is always a debate regarding which more important.
Profit Maximization
The basic concept behind profit maximization is to earn a large amount of profit. It is a short term objective of the company (every fiscal cycle). There is no consideration for risks and uncertainty. It acts as measure for operational efficiency of the company. Profit maximization is necessary for growth and survival of the company.
Wealth Maximization
The goal of wealth maximization is to improve market value of shares. The main focus is on achieving long term objectives. There is consideration for risks and uncertainty. It tends to gain a large market share. It accelerates the growth rate of a company.
I cannot say which one is better. Profit is basic requirement for any company. Profit feeds oxygen in the system to take breathe and survive. As we know profit is directly proportional to the risk. Higher the profit, the higher will be the risk. Risk factor can be neglected for a short run (profit maximization) but in long run (wealth maximization) risks or uncertainty cannot be ignored. Shareholders invest in company in hope of getting good returns out of it. If they won’t get returns then chances are that they lose their trust in company and may sell the shares, and invest somewhere else. All this effects reputation of a company and ultimately the market value of shares fall down.
So, I feel profit maximization is ok in decision making. But when some decisions affect the interest of the shareholders the wealth maximization should come in place.

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When somebody starts a business, getting high profit out of it is the sole idea. Companies became more obsessed with maximizing quarterly/yearly profits. Giving more efforts on increasing sales and trying to reduce operation cost, giving less wages to employee. Overall became more thrifty. Profit becomes the source of finance. I believe sometimes profit maximization may lead to employee/customer dissatisfaction. And sometimes to get profits, companies tend to take bigger risks.
Wealth Maximization I believe is like taking care and increasing value of all, for eg. Customers, workers, shareholders. Wealth maximization contributes economic growth of a country. Here profit becomes the long term goal.
Let’s have an example:
Company A (not having good pay-out policies), working on sole intention of maximizing profit, pays out less salary to their employees. Which leads less spending power of these people, if profit is aim of maximum companies of that country then it can tend to slow down overall economy. As maximum people will be earning average or just above poverty line.
For example in America, Walmart employees, spend everything they earn on buying food, clothes, gas, houses, entertainment and other products/services. This money spent by people becomes profit for companies which provide these product and services.
As per facts (from businessInsider Dec 2012), Walmart employs 1.4 million Americans, Average full time associate makes $12 an hour--$480 a week and $25,000 a year. That's just above the poverty line. So these people are nearly poor. Walmart made $27 billion profit (2011).
Moving to Wealth Maximization: If Walmart gives salary raise to its associates, then it may lead to more customer satisfaction. Increase loyalty, getting better employees, reduce training\hiring costs (agree that it will come with reduced profit of companies).
By paying associates more, companies will also put more money in hands of consumers, so increasing their spending power, so this way companies will help in accelerate the economy growth. And if economy will grow, the companies will grow.

A lot of profit maximization and wealth maximization definitions, explanations and differences are shared. so , I want to further go deep with an important and difficult problem knows as Agency Problem.
On the path of wealth maximization big companies faces Agency problem. The agency problem arises when one person hires and authorizes another person (agent) to act on his/her behalf
In big companies, Managers are appointed by shareholders(owners) to make decisions. They expect these decisions to be of shareholders interest, but the interest of shareholders and managers are interwoven.
The managers have the control and it is possible that they may put their personal interests first rather than the interests of the company and its owners. This situation leads to conflict of interest between the shareholders (the owners) and the managers of the company. This problem is known as the Agency Problem.
Example 1: The company is considering a new investment that is risky, but can contribute to increasing shareholder value. The investment is in the interest of shareholders but not to managers who may lose their jobs if the activity does not take place as planned. Precisely because of the existence of different objectives managers may decide not to implement this kind of investment even though it would bring harm to the owners of the company.
Example 2: The managers can be against the decision of merging of their company with another one although that would be in the best of interest of the stockholders. This behavior of the top managers might be a result of the possible and very probable change of the leading managerial positions after any merge.
Controlling the Agency Problem: * Internal audit * Change in the salaries and payments of the managers * Concentrate ownership * Good corporate governance/management.