College of Engineering, Pune

COEP is a premier institute while SPCE is not even known by that many people. Among the two COEP is the winner. COEP has been ranked one in Maharashtra after the IITs and NITs. According to MHRD rankings COEP is ranked 24th while SPCE is on not even in top 100.
Generally speaking COEP blows SPCE out of the water. COEP is much well-known and trusted among recruiters as compared to MIT. Last time COEP had around 83.8 % placements for mechanical (this number seems less as the students who didn't accept the offers are also included) and the average salary was around 6-7 LPA for mechanical department which is more compared to SPCE.
COEP in also better in academics and extra-curricular are compared to SPCE. What I have learnt from staying one year in Tamil Nadu is that COEP is well known all over India but the same is not the case with SPCE.
So go for COEP over SPCE.

Getting placement abroad is not a very big task nowadays where more than a thousand companies visit India from across the world to extract the most creative and passionate minds. It strictly depends on you that how you are taking your studies and performing into the screening and interviews.
Now when it comes to COEP, it is one of the oldest colleges affiliated to Savitri Bai Phule university in Pune and is among the most renowned and shining colleges of India. When it comes to getting placed abroad, there are many multinational corporate and industries visiting in COEP for the recruitment drives and many students directly get placed abroad with a very high bank balance and quite good privileges to them.
For getting placed abroad, keep in mind to get a core branch which will increase your chance to get your desired career. The most number of students are placed from IT, CSE and from EC branch. Also, you can work in India for one year and if company find you an asset, it will directly transfer you to their abroad headquarters. Hope it helps.

Yes, there is a quota of JEE Mains in the state colleges of Maharashtra though it is preferable to give MHT-CET for getting into the state government colleges of Maharashtra. If you are seeking admission into COEP you don't have to worry about the branch. Above 200 marks will be sufficient for you.
If you want a core branch such as computer science and engineering, mechanical engineering, electrical
engineering and so on you need marks above 250.
The above cases are for the general student. For an SC and ST student, the cutoff is very low as you do not need to touch even 200 for getting into the CEP as a core streamer.
Same for OBC the cutoff is lower but slightly higher than SC and ST which can be above or around 200 but still have more chances of getting into COEP than a general student.
Now in my personal advice, if you are very much worried and focused about placements, then go for petroleum in COEP. It will make you a millionaire in a very short time but not without hard work.