Are MD and Partner the same?

for the most part no, they are not the same.

Depends on the firm. At some, all the MDs are "partners"

Case by case, but partners typically share in equity unlike more junior MDs. Partners are often part of "MD," but not all MDs are Partners.

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Typically MD is the level just below Partner in IB. In public banks, Partners have access to a dedicated bonus pool reserved for solely Partners. In small firms or private ones, Partners may also be given some equity in the company so they have direct skin in the game. The latter is the more common model in consulting. 

At Evercore, MD is the position right after VP which I understand is called "ED" at other banks. There's no official "Partner" title but everyone who is an SMD (Senior Managing Director) is called Partner in internal but not official communications and daily speak.

There are MDs who are 100% execution-focused and are the right hand of successful SMDs and there are other MDs who since Senior VP start developing coverage and clients to make it to SMD in a short period of time. That's why there are a few SMDs who did only a few MD years (Naveen?) and other MDs who have been MDs for a while without a clear path to SMD. Different strokes for different folks.

MDs at RBC need to buy in equity so they are effectively like Partners despite no title as such

It depends on the institution. In the past, a partner was always somebody who held equity in the firm and therefore received a share of the profits above what would normally be given as salary and bonus. Nowadays, a partner is typically an MD who has performed very well, but they do not necessarily have to hold equity. For example, partners at Rothschild & Co hold equity and receive profit share, but partners at Goldman Sachs do not; however, partners at GS do have much higher bonuses and a starting salary of I think $1mm. It is not common for publicly traded companies to have equity partners because of how diluted their stakes would be; the example I provided of Rothschild is different because the float is only a minority of shares outstanding. Some other companies do allow for profit sharing, but not in a partnership format. One example of this is at Baird, which has employee stock purchase plans (ESOPs). Some places can choose to have both equity and non-equity partners, but this is more common in law. For example, Kirkland & Ellis has both types whereas Cravath only has equity partners. In any case, someone with the title of "managing director" typically does not hold equity at any sizable firm. In addition, true equity partners typically have a say in the major decisions in the company which would be enforceable by the company's bylaws, while an MD would not. Using my previous example of Rothschild vs GS, that would mean partners at Rothschild can actually force the company to do certain things because they hold equity, but partners at GS cannot as a function of their title as partner.